catching up on this symmetric difference stuff. lots of great ideas there.

. o O ( circumsection? )

Shame on you for not reading the Hugh Lofting books as a kidlet, @robjohn.
Did your client enjoy his hours of getting yelled at, @Leslie?

What's wrong with the song?

I don't know the movies/songs.

3:12 AM
lots of ways to prove associativity of the symmetric difference, i think it's conceptually cleanest to think in terms of indicator functions and addition mod 2, assuming you know that's associative.

Ah, that's the song they sing when they see the pushmi-pullyu

That sounds pretty fancy schmancy.

he did enjoy it. i think he had too much fun.

Oh, OK ... I only read books.
I was disappointed that Xander had better options for his worksheet question.

my cat appeared on zoom at one point.

3:14 AM
aren't the zoom days about over? Superior Court here has been in person for months and months. I delayed jury duty but am on in about a month.

a lot of depositions are still virtual. maybe they'll just be virtual from now on. it was for a proceeding at the patent office where the judges don't even look at video, it's transcript only. i could see wanting it in person if the person is on tape.

I don't think Zoom will be going away. The option of Zoom meetings, etc. is too wonderful.

Well, SD courts are in person as of many months.

a number of federal courts have started up again too.

In LA, they opened in January, but the number of cases were very limited.

3:20 AM
just the people of the state of california versus robjohn, in the small matter of the armed robbery of that delivery truck for $200 in consumer electronics. Gotta have enough joy buzzers around the house! one time my advisor made a cryptic remark about having to cancel class because he had to be in court, and someone asked 'oh jury duty?' and he said 'no' and did not elaborate. i think he was a witness to something, but the joke after that time was that he had carjacked a shipment of DVD players. i forget where i came up with that. just seemed like something a math professor might do on their off time. so a multivariable function can have directional derivatives, not be continuous, and not be differentiable then?....just deducing some ideas based on an exercise I just did also Good evening to the the crew.... yeah. No quick witted story on directional derivatives to follow?..........you must be tired from chasing the lil one around today 3:35 AM directional derivatives are just information along very specific paths. without some slightly stronger assumption about regularity the existence of directional derivatives is not going to get you much ah...this is the Leslie I expected consider f(x,y) = 1 if y = x^2 and x is nonzero, and f = 0 otherwise. so you have this parabola arc getting right up to (0,0) in this goofy way but you won't see it if you look along straight lines. any straight line is going either not to cut the parabola at all, or cut it at some distance from zero. maybe not the best example. you can come up with rational functions defined slightly less artificially. maybe that's what you were looking at. was working with$\frac{xy^{6}}{x^{4} + y^{8}}$when I did the question, but I played with some examples along different paths.....very subtle and finnicky stuff something can be fine along rays without it meaning that the behavior along different rays will relate very well, or that you might not be fine if you aren't approaching via rays. There are discontinuous functions that are continuous along every$|y|=c|x|^n$,$c\ge 0$,$n\in \Bbb N$. 3:51 AM interesting. i'm guessing something like my model above but with a smooth bump function in the role of y = x^2. no pictures. i only see this from algebra. I was just spending the last 7 mins looking at the paths that Ted was describing with regards to possible surfaces where this occurs 4:09 AM It's generalizing a few exercises in my book. 4:35 AM humans are too sensitive they get offended by anything 1 hour later… 5:44 AM what do you mean by that???? how dare you say that 5:57 AM human should be huperson, wait, it should be huperoffspring. where's my apology copperhat with the late 80s material. i'm a 90s guy, copper. we don't do that kind of joke anymore. my son graduated from high school today, as if i needed another reminder of my 'experience' my daughter calls me fossil. just in case i wasn't getting the general drift... just wiser.....amor fati is the outlook to have. now I do have a computational question to ask.... 0 Show for any$a \neq 0 $we have: $$\lim_{h \to 0} \frac{\|a+h\| - \bigg(\|a\| + \frac{a\cdot h}{\|a\|}\bigg)}{\|h\|} = 0$$ Note: all terms are vectors of any dimension. Hint: remember that$x - y = \frac{x^{2} - y ^{2}}{x+y}$Attempt: I've gotten this far $$\frac{\|a+h\|^{2} - \bigg(\|a\| + \fra... 6:25 AM The derivative of \phi(a)=\|a\|^2 is D \phi(a)h = a \cdot h. The derivative of s(y) = \sqrt{y} is {1 \over 2} {1 \over \sqrt{y}}. \|a\| = s(\phi(a)). Just from examining the definition of what the derivative is I was able to deduce that, actually that is what the second part of the question asks me to find...the derivative congratulations on your son's graduation, copper Thanks Lesiie! @dc3rd The derivative of \phi (pronounced fee :-)) is easy to compute, as is the derivative of s. You can compute the derivate using the composition rule. Ok. THat makes sense. But how do I arrive at the conclusion from the first principles as I have asked above? namely showing all of that stuff equals 0 ? in a few weeks my daughter is going to 'graduate' from what they refer to as 'toddler' day care to 'primary' day care, which is a mix of people from 3-5. she is only about 2 and a half. on one level it's just babies in a room but on another they do assess emotional and cognitive development at her day care, so seeing her get in 'early' feels good. 6:30 AM Exciting times :-) Fun when you can still lift them up. those days are soon going to be behind me. my daughter was always large and i do not have upper body strength. she still does sometime ask me to carry her up the stairs. i will do it until it feels dangerous. @dc3rd I suspect the first principles may involve essentially the same sort of computation I have above. even when i was in good shape, the torque of my kids off to one side tweaked my back. sometimes when she is misbehaving and we have to carry her away from the beloved duck pond there is quite a lot of distance to travel. i just throw her over one shoulder. far better for my back. i've noticed other parents don't do this. jewel lake was too far for gymnastics, always a stroller. you're definitely gonna wanna write as much of that as possible in terms of the dot product. hence the hint about squaring. 6:41 AM jewel lake is almost gone :-( is it? that's sad, but not surprising. i haven't been in ages. does lake anza still exist? i used to house-sit next to it. there copper I corrected the mistake in my comment :-) lake anza is still there, but no swimming due to algae content, apparently. @dc3rd i added the expansion needed using the trick you wanted. Thanks. It provides completeness for the question, I was just writing it out myself on my notepad just now too i wouldn't swim in anything these days except maybe a highly chlorinated pool. water is filthy. we all know what fish do in it. 6:49 AM i am at home in filth having spent countless hours in the bay mostly while (not) windsurfing also kayaking, and sometimes swimming (when thrown off a boat) my wife once fell into the iowa river while rowing. she told me about it and i laughed, which was not the right response. @dc3rd it is worth looking at the connection between the formula i have above with the expansion in the answer. why, is the iowa river polluted? read my mind.... i think the worst water i have been in was (i) the outflow off san mateo during heavy rain and (ii) haikou bay. not any more than any other river. but she capsized next to members of the iowa rowing team, which she was not a member of, and who didn't help her. and that's pure comedy from my point of view. it wouldn't be funny if it happened to me. 6:53 AM b*stards i got a big digestive disturbance from the san mateo outflow i learned subsequently that in heavy rain, the sewage goes more or less straight to the outflow. it is a mile of so out, but still, how was i to know. one time in the dead of winter i saw some tracks across the frozen surface and decided to risk it instead of taking a bridge. where were you then so i could take action? there's a spot near here that is like that. has the statute of limitations run? :) on foot or in a vehicle? on foot. it was really, really frozen. not as risky in retrospect as it felt at the time. 6:56 AM well, hindsight and all that. @dc3rd are you able to read deleted answers? no, not yet....should I screen grab your answer? no hurry. i'll delete it tomorrow or sometime after that its part of a cunning perverse rep hunting plan i have lol....I have screen shot it already...two clicks or are you hunting your mystery downvoter? i'm off to cotton island no, some user was accusing me of hunting rep recently :-). Lol....did they not glance at the number beside your 8 gold awards as if you are in need of rep..... 7:02 AM i hunt rep. i can't get enough of it. me on the other hand. I'm still basking in the pride of getting my first answer when I explained the central limit theorem to a poster well, i was happy to get my mug when i hit 100k. it was the reward for countless hours of procrastinating. actually, some of my early answers are a bit embarrassing. you should try explaining it to a human being takes a while to get back into the swing of things. maybe it was a poster child? my answers are a long steady decline in quality. the earliest stuff is the least embarrassing. 7:04 AM one who had lost their iid? now it's just, "nurse, he's out of bed again" for me its the big red button now before my browser crashed I was going to say " I aspire to get to a level of proficiency where I can look at math problems as being "procrastination" from my regular work" its a bit depressing to read some of my answers and realise the enormous cognitive decline that has occurred since i wrote it... i kinda miss math being my regular work. it's pure procrastination now. well, that and dunking on high school students who have far less training than i do. just dunking, left and right. it makes me feel big. 7:07 AM i liked numerical work, but it was typically 5% interesting analysis and 95% grind at the kb and one is always working on a flea's worth of an elephant problem, so ultimately not satisfying. on the other hand, product dev actually gets some stuff done. another little boundary condition of some FEM code handled. whoooeyyyy i just gotta handle a mortgage and day care until my daughter can become a ward of the state. much more satisfying when a happy customer moves on to a 3m order i am hoping to become a ward of my daughter i'm trying to get her involved in the entertainment industry. i understand that only good things happen in it. if you're working on a flea's worth of the problem, who get's to "put it all together" and get the satisfaction ? :-). what was your comment about huperoffspring? 7:12 AM did you mean thing$$$ Leslie?

@dc3rd i mean when i work on the numerical aspects of solving something, it is like working on the design of some part of an engine whereas the product dev is more like the whole car.

i do know several people who are very comfortable because of relatively small and lucky and well-timed amounts of work in the industry. nice lotto to win.
maybe not the nicest lotto to play.

i have nothing against pay for play, but the whole industry is just rife with exploitation.
personally i think pay for play would solve sooo many modern first world problems
@dc3rd when i worked on optimisation algos in industry i had maybe 1-2 customers who loved my stuff

one of my friends had a roommate whose job was playing a corpse on one CSI or another for about 10 years. precarious work and a surprising lot of time sitting around just to be dead somewhere.

honest pay for play....because the exploitation is what happens at the moment. Get a new set of developers...burn them out with work, rinse and repeat..

7:16 AM
@dc3rd more recently i developed code for a company whose name starts with G and they were very happy and it morphed into a big order.

ahh, GloboChem.

much more satisfying even if i don't participate in the %

my favorite.

I always get irked because the people behind the product don't get to enjoy in the true royalties of their art piece

i just do code dev now. i'm tired of insurance, negotiating, silliness, etc
@dc3rd well, i thought that until i had my own company. it is a lot of grind.

7:17 AM
in this case the other Alphabet boys get to enjoy the splendors
guess when I experience running my own firm then I'll see the other side of the coin.

risk takers need to be rewarded, i am taking little relevant risk.

i've worked with a lot of biotech startups that do a pretty good business in not asking for participation and doing very competent but non speculative work.

that's true

i was rewarded when i took risks

and if some idiot wants to overpay for a half baked hamburger because they see possibilities, so much the better for them.

7:19 AM
that's another story
its a sort of ponzi scheme

i have some Leslie Technologies that i'm happy to sell. no percentage required, just money to me up front.

well I want my payout in LeslieCoin then

LeslieCoin is a groundbreaking advance in people giving money to other people. it's never been done in quite this way before. bitcoin is like version 0.0 of what lesliecoin is.
people keep talking about it. that's how you know it's real.

g7 will come looking for you

criminals are not using lesliecoin. the only people using it are great people who are happy to trade antiquated forms of "money" for something of true value.

7:29 AM
g7 does not care about criminals, they just want 'their' 15%

i do wonder what some of this will do for some of my favorite tax jurisdictions.

i am voting for an irexit
on that note, good night :-)

goodnight.

night

i'm at that point where i need to decide, am i really growing a beard or not.
harder than most math problems.

7:43 AM
hi maths
My tutoring session was postponed, to today, and I just got back
The kid was doing integrals and I think I failed to fully channel Ted (though, I did remember to invite him to least start by drawing a graph)
Also I was massivley distracted by the prominent copy of mein kampf sitting on his otherwise desolate bookshelf

whaaat

I know right
I hope he was studying history

i'm torn between, don't tutor him, tutor him on something other than math, or just take all of his money.

There is scope for life lessons in tutoring maths right
although I'm not sure I will be asked to come back, I couldn't help him solve what seamed like a trivial problem

depending on the subfield, i think so. a lot of the most basic stuff was worked out by fairly wealthy people in fairly privileged conditions a long time ago. i don't know what the lesson is there except it's good to be rich in the 17th century, or whatever.

7:48 AM
(had to find the area under a curve between a cubic and two straight lines... and his calculator kept spitting out the "wrong" answer despite me conviced of the algebra)

more recently there is stuff on both sides of the ledger. mathematicians who were prominent nazis or fell into line with mao or whatever else, and mathematicians on the other side.

I'm feeling bad for judging someone on their bookshelf. I had some pretty shitty books on my shelf at his age
(a lot about how aliens put stonehenge in place as a teleportation platform for seeding life blah blah blah)
Also James Watsons biography

banach of banach space fame was used by the nazis as a breeder of lice. they were testing some kind of typhus thing and needed people for lice to feed on to do their experiments. so he just provided blood for lice. which tells you exactly what it was to be a nazi.

the fact that I couldn't solve this simple problem though is giving me pause as to weather I actually make a decent tutor

sometimes it's more about asking questions than having every answer. in any subject, and i'd go right down to basic algebra with this, there are calculations that are not intuitive and do require levels of thought and perseverance that a tutor might not have immediate access to.
thinking about a thought process out loud, providing strategies for approaching unfamiliar material, you might have experience in that even if you don't know if the integral evaluates to 0 or not.
that kind of thing.

7:55 AM
I hope that my thinking out loud helped
I took a notepad and kept drawing the problem out
and each time refining the sketch as we got closer to the solution

multivar calc is a good example of this. a lot of textbooks require bit of computation to solve problems. you can fill half a page, sometimes, if you organize your work carefully and do not try to do everything on one line.
knowing what you might try to do, and why you might try to do it, is far more important than being able to fill that amount of paper with the calculation that gets to the result.

in fact I got so stuck into it, his father had to remind me I had overstayed my time by 15 minutes, and asked, politly, if I was planning to leave

i'm notoriously horrific at long computations. i can't diagonalize a 3x3 matrix, i can't take a surface integral without double and triple checking. but i do have a sense of the structure of what the subject is about. that has value.
"sir, are you planning to leave" is never a good moment :)

I know right

i don't know if it's the same in australia but in the US if you're in a store and someone calls you "sir" you know it's about to go badly for you.
there's going to be some kind of request for you to leave or explain why you are there.

7:58 AM
Simply having someone say "Sir", or "Mam" is usually enough
You often don't need to follow up with specifics, in Australia being polite is always code for "fuck off"

in other countries, i think it works differently. someone told me in some parts of india it's actually used as a sign of respect, or just a social default, and not "please get out of here"
the southern US is particularly bad about this. you can tell how rude someone wants to be to you by how polite they are being to you.

@TedShifrin I couldn't bring myself to steal your shtick. Instead I kept saying : "My friend Ted would probably be asking...."
@leslietownes thats that whole honour code malarky
I'm reading a great book atm called "See what you made me do" (depressing, but super interesting). The author talks about how people use politeness as a threat to preserve their self image. It allows them to intimidate others, by what is implied to happen if you don't comply, while saying to themselves "I'm not violent, I'm a polite person"

i grew up in a slightly different culture. my parents were both raised in households where you expressed familiarity and even affection by being very clear about what you didn't like. no guessing games.
we're friends, we're family, we can skip the social norms and just get to me telling you off.
kind of the vibe i internalized growing up.
my daughter first began using the f-word in contextually appropriate ways, as in "f--- going downstairs, i want to keep playing in my bedroom," around age 2.
she has inherited it.
the toxic legacy.

I don't see the big deal so long as it is contextually appropriate

exactly. i don't want gratuitous use of the word.
if it makes sense in context i barely even hear it.

8:07 AM
haha.
Ahh looks like my boss wants a debreif

tell him that f--- going downstairs and that you want to keep playing in your bedroom.
worked for my daughter, anyway

lol, She is a very nice person. I mean she offered a tutoring gig to a mechanic
:P

and now i'm a monster for assuming a boss was a 'him.'

yup
patriarchy

i mean, most of them are. if you play the odds.
but not for any good reason

8:11 AM
but now she isnt happy that I didnt sell another session :(

i've got a light on on my honda, maybe you could help me with that? would it help her?
i actually should get to a mechanic, i've driven about 50 miles in the last year, so it's hard to see the demand for it. but it hasn't been in for a checkup in a very long time

what sort of light?
is it a orrange/yellow exclaimation in a triangle?
thats a bad one to have on

just a low tire pressure thing, which is why i'm not worried about it. i know which tire, and it's not too under inflated. it's not one of the spooky ones.

mathematicians and mechanics are similar in that people often don't want to talk to them until they really want to talk to them.

8:16 AM

i mean i could but the nearest service station is a good 5 minute drive from here.
so you see my quandary.

oh, lockdown or lazyness?

a little of both.

fair
Sir, I really think you should drive to the service station and inflate your tyres

i should, i'm just beyond dumb about everything.
i'll pay in lesliecoin.

8:18 AM
That probably makes us even seeeing as I owe you lesliecoins already
(and given the clear depreciation in the value since our last transaction)

yes, we are even. although, one can never have too many lesliecoins. it's good to maintain a surplus.
you might call it a strategic reserve.
sometimes people come to me in tears saying that they wish they had bought more, and i say, don't despair, there is still time.

That is how I feel about every exponentially growing stock I have become aware of after the fact.
(I only ever seam to catch them in the decline phase)

because of my job i basically can't hold stock. i can only hold index funds. this reduces my blood pressure considerably. i couldn't even buy stuff before it goes up.

that sounds like enforced wisdom
night

enforced wisdom is the best kind of wisdom. night.

9:20 AM
Another question to American native speakers: semi-simple or semisimple? (about a Lie algebra)

i think the latter is fairly common.
either one.
putting up the bat signal for actual users of lie algebras.
i think the absence of the hyphen feels more 'modern' than putting the hyphen in, but am not a specialist.

I’m trying to prove order 15 group cyclic.
If all elements (non identity) have order 3, then I’m not getting any contradiction.
For if every non identity element has order 3 then no of non identity elements should be a multiple of $\phi (3)$, which is fine here as we have 14 non identity elements.

3 hours later…
12:38 PM
@TedShifrin Hey Ted! were you able to ponder over it? I thought about it and it seems to work for curves with curvature as well such as $ds^2 = (1+y^2) dx^2 + (1+x^2) dy^2$

2 hours later…
2:12 PM
@TedShifrin diff geo with Cheeger or probability limit theorems with some French dude? I love both subjects, and the tradeoff is that probability is the more difficult class, but diff geo is taught by Cheeger (but the word is that it's an easy A, which turns me off). I have a feeling I know which you prefer..

2:49 PM
@AndrewMicallef Oh, I still have books like that on my shelves. They are great when you need a laugh.
@AndrewMicallef I think that it is too much to ask that one be able to solve every single problem on the fly. Particularly when working with integrals, where there are a lot of ad hoc techniques which one has to recall on the fly.
If you haven't been immersed in it recently, there may be little hope.
Rather, the tutor should be able to help the student think about the problem.
And should be able to quickly find appropriate references to help out.

3:08 PM
In the first table in the linked notes, what exactly is the definition of "order of magnitude estimate"? How can $f(s) << g(s)$ and $g(s) << f(s)$ for $s\in S$ when this notation is previously defined as being equivalent to the Big oh notation, where either $|f(s)|\leq c|g(s)|$ or $|g(s)|\leq c|f(s)|$ for all $s\in S$ and some constant $c$? faculty.math.illinois.edu/~hildebr/595ama/ama-ch2.pdf

3:18 PM
Also, in the definition of the Omega estimate, what limit is there left to take when one has taken the supremum of $|f(x)/g(x)|$ for $s\in S$? Will this supremum not be a fixed number, that is, a constant?

@schn Protip: the commands \ll and \gg render as $\ll$ and $\gg$. :P

@XanderHenderson Good protip :)

3:34 PM
@schn So the notation $f \ll g$ mean, per the definition given in the linked .pdf, that there exists some constant $C$ such that $|f(x)| < C |g(x)|$ if $|x|$ is large enough (i.e. $f \in O(g)$).
This constant gives a lot of wiggle room.
Consider, for example, $f(x) = x$ and $g(x) = 2x$.
$f(x) < 1 \cdot g(x)$ for all $x > 0$, so $f \ll g$.
On the other hand $g(x) < 3 \cdot f(x)$ for all $x > 0$, so $g \ll f$.
@schn $\limsup$ is a single operation.

A high school calc book that I use says that we could take the limit inside the integral when the bounds are constant. Is this true? All I could find is dominated convergence theorem which I couldnt understand(calc 2 moment). Is this just an oversimplified version of that theorem?

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant...
@DatBoi I don't understand what this means... which bounds?

bounds of the integral
sorry if that wasnt clear

Do you mean the upper and lower limits of integration? or bounds on the integrand? or bounds on the value $\int_{c}^{x} f(t)\,\mathrm{d}t$ as $x$ varies?
Which bounds?

the upper and lower limits of integration

3:41 PM
it probably is just an oversimplified version of DCT

@DatBoi By "constant", do you mean "not infinity"?

let me give an example
For say, $\lim_{x\to0}\int_0 ^2 \sin(x)dx=\int_0 ^2\lim_{x\to0} \sin(x)dx$
0 and 2 are constants here because they are not dependent on x

but you're also taking the limit of a constant (w.r.t x)

I am confused.
You have $x$ doing two things there.
It is the dummy variable of integration, and you are also taking a limit with respect to $x$.

are you sure you don't mean something like $\lim_{n \rightarrow 0} \int_{0}^2 sin(nx) dx$?

3:45 PM
In any event, the (uniform) continuity of $\sin$ is important here.

omg im so sorry
xander is right

Well... maybe not... it depends on what you mean by the notation.
I'm confused.

@porridgemathematics this one

if you're working with a riemann integral, uniform continuity is probably what you are using to interchange the order of the limit/integral
that being said, in this case its enough to just apply DCT, but as you say you can't use that yet, so its probably the uniform continuity thing

5

Let's say I have a (uniformly) continuous functions $f:[a,b] \to \mathbb{R}$ and an arbitrary function $h:\mathbb{R}^2 \to [a,b]$ such that $$\lim_{t\to 0} ~h(t,x) = h(0,x) = x$$ for all $x$. I would like to be able to conclude that $$\lim_{t\to 0}\int_a^bf(h(t,x))dx = \int_a^bf(x)dx.$$ ...

@porridgemathematics My very vague recollection is that there is a reasonable proof of "uniform continuity let's us move limits about willy-nilly" using the DCT.
DCT is the greatest thing ever.

3:49 PM
@XanderHenderson it totally is :) I think I found 'unlearning' the riemann integral a little too fun learning about lebesgue integration the first time

as for the uniform continuity thing being implied by DCT, let $f_n$ be defined on a compact interval $[a,b]$, such that $f_n$ converges uniformly to $f$ on $[a,b]$, then its easy to see $f_n$ is uniformly bounded in $n$, apply DCT, profit
(everything here being riemann integrable)

ah yes
the post is truly enlightening
actually this came up when i was looking at this post
1

Fix any $a$ in $(0, \pi)$. For each $k$ in $\mathbb{N}$ define the sequence $s_{k}=\int_{a}^{\pi}(\sin k x) / k x d x$. Prove that $\lim _{k \rightarrow \infty} s_{k}=0$ In order to tackle such exercises, we have to find some $\delta$ for each $\epsilon$ in which whenever $k \geq \delta$ holds, t...

just realized I think 'uniform continuity' in some previous remarks should be changed to 'uniform convergence' @DatBoi
this is likely what was being meant anyway

cool, i get it
thank you all

4:14 PM
Hey if any1 can help with a little bit of representation theory of finite groups would be cool
0

Let $N \unlhd G$ and suppose $G\diagup N$ is abelian. Let C be the group of linear characters of $G\diagup N$ so that C acts on $Irr(G)$ by multiplication. Let $θ \in Irr(N)$. Show that $θ^G=f \sum x_i$ where $f$ is integer and $x_i \in Irr(G)$ Try: I tried using gallaghers theorem but co...

4:38 PM
@XanderHenderson Thanks for the reply. You wrote "...$|f(x)| < C |g(x)|$ if $|x|$ is large enough...". Did you mean $\leq$ and why "...if $|x|$ is large enough..."?