no, no, I mean try doing $\theta = 2\pi k/n$, $k=1,\dots,n-1$ for the location of $B$. And then for each one, what's the likelihood of a $C$ that works? Try to work it out.
Some of my teachers make a big effort to explain why things work and why we need them. Others just hand out cheat sheets and say "memorize this for test".
@Bob I'm going to self-study the concepts for the class before I get there so that it will be second exposure for things I don't understand. That's what I did for the calculus class I'm in now.
And I think it's okay to some extent. Not everyone has to learn mathematics. Just like you can't expect everyone to learn to play an instrument or learn to draw well.
But it's O.K. they teach it this way because I know if they taught it the way that I'd want them to, then there would likely be serious deficiencies in "symbol pushing skills" in engineers, scientists, social scientists, etc.
Hi guys, taking a real analysis course; is it sufficient to use induction to prove that something holds for all natural numbers in order to prove that it is true as n goes to infinity?
I see... I'm trying to prove that $lim_{N \to \infty}sup \{a_n:n\geq N\} \geq lim_{N \to \infty}inf \{a_n:n\geq N\}$ for some sequence $\{a_n\}_{n \geq 0}$
in this case you can indeed reduce to the finite case because it is $\le$ instead of $<$. But this is a non-trivial theorem that you should first formulate and then prove.
I'm a bit confused; why is $\leq$ vs. $<$ a significant difference?
I wanted to give an argument that's similar to:
denote $ x_N := sup{a_n : n \geq N} $ and $ y_N := inf{a_n : n \geq N} $, and give a proof like the following:
$x_N \geq y_N$ for all $N$
or, use the property of sets and just claim that the infimum of a set and the supremum of a set will always be the upper/lower bounds and so the $sup \geq inf$ is true in the limit (but ofc in a more formal way than what I just described)
@MaryStar Sorry I was asleep, I saw that JoeShmo got you covered and you also know you can prove the contrapositive if you want something more than a conterexample
@Secret how do hell do you do proof by contrapositive to a false statement? Counterexample is great and their idea for how to construct the counterexample is also great
in this case you can indeed reduce to the finite case because it is $\le$ instead of $<$. But this is a non-trivial theorem that you should first formulate and then prove.
do the former first.
what does it mean that "you can reduce to the finite case"? What is the general theorem?
@LeakyNun I guess my question wasn't clear; I meant in that quote, "reduce to the finite case because it is ≤ instead of <.", what you meant by "≤ instead of <". As in which inequality you were talking about
@Holo Ah, clearly, my brain is not functioning, I forgot that the starting implication is false thus its contrapositive is also false. Sorry about the confusion
I also not just satisfied at finding a counterexample, I will try (if it is feasible) to find the whole set of counterexamples, thus finding counterexamples are really finding a set of examples obeying a proposition that is required for a given counterexample
My favorite approach in finding counterexamples whenever I am not familiar enough with the mathematical object $M$ is a top down approach.
Start with the most general example of $M$, and the find the condition that all counterexamples of M has to violate, then pick one concrete counterexample fr...
I use this technique a lot when constructing highly pathological objects
the answer to my question is that the general theorem would be that if $x_n$ and $y_n$ are two convergent sequences such that $x_n \le y_n$ for all $n$, then $\lim_{n\to\infty} x_n \le \lim_{n\to\infty} y_n$.
There is no amorphous set of reals - in fact, no amorphous set can be linearly ordered.
(For further discussion of what kind of structure amorphous sets, and more generally Dedekind-finite sets can have, see this paper of Truss or Agatha Walczak-Typke's Ph.D. thesis.)
To see this, suppose $A$ i...
I am not ready on amorphous sets yet, the chemistry keep pulling me around so I don't have much time recently to deal with set theory stuff which is why I am mostly lurking in the chat
Suppose $x_n \to x$ and $y_n \to y$, and for the sake of contradiction, $y < x$. Then by definition, there exists $\epsilon > 0$ such that $|x_n - x| < \epsilon$ for all $n \geq N_{\epsilon}$, and similarly for y
and then pick $N_{\epsilon} = max(N^x_{\epsilon}, N^y_{\epsilon})$
where $N^x_{\epsilon}$ denotes the N such that all $x_n$ is within $\epsilon$ of x, and similarly for y
then you can pick any element past $N_{\epsilon}$ in x and y that contradicts the original statement, and so we proved that it must be $\leq$ and not $>$
amorphous sets proofs are interesting because it helps to teach us how to reason about partitions at a very fundamental level without the considerations in nice orderings
The terminology "normal" is ubiquitous in mathematics. The only commonality I can find is that the objects involved does not in a way "depends on each other"
I can easily check whether someone has ignored me by probing the continuity of the chat flow
I can also do that check between A and B by checking whether their conversation flows
It's a bit hard for people to apply this onto me because I do have instances where I generate a series of monologues that is directed to no particular people
though this is a lot less common now because the chemistry will often pull my soul elsewhere
Well, just watching conversation flow doesn't always paint the whole picture. For example, I have been in chats before where I just decided to change the topic I was talking about at the moment solely because another thought occurred to me.
Does anyone know of any online writing app (something like pasteOfcode for programming codes) to write Latex math notations to be suitable for reading?
Hmm, doesn't that only contain features for like writing books or personal projects? It's not really suitable for sharing on casual platforms like these with just the sending of a link of some sort?
I'm trying to help my 12th grader with a math problem. Given a real $a$ such that $a^3+a^2 = 1$ (hence in $(0,1)$), compute the infinite product $(1+a)(1+a^2)(1+a^4)(1+a^8)...$. I've been struggling for a while and am not seeing the light. Any obvious tricks?
It's one option for sure. Peter has told me about some places to look at (though his suggestions were fairly ambitious so I should prob ask for some places that are more safe-ish), though he hasn't mentioned too many specific names of folk except for Mike Hill at UCLA
AG is real nice so far and the vibe I get from the more advanced bits is that it intersects almost everything, AT (also possibly other types of topology) is something that I think I'd like, then NT/algebra/rep theory, tbh even stuff along the lines of functional analysis wouldn't be out of the picture
Hi, sorry for the handwriting if it's not legible, my tutor tried proving part (c) by contrapositive approach and I don't understand what he's doing. Does anyone understand it?
e.g. take $\mathbb{P}^1$, with the usual charts (so the transition map sends $z\mapsto 1/z$ )
So if you consider the function $f(z) = z$ on $\mathbb{P}^1 = \mathbb{C} \cup \{\infty\}$ (so here implicitly I already took a chart) , if you look at this function on the other chart with coordinate $w$ it looks like $f(w) = \frac{1}{w}$
so you see there's one zero and one pole
On the other hand if you look at the meromorphic 1-form $zdz$, this has a zero at $0$. On the other chart, this looks like $(1/w) d(1/w) = -1/w^3 dw$, so there's a pole of order $3$ at $0$ on the other chart (in other wo…
@LeakyNun i see that $0\le \sum\frac{\sqrt{a_n}}n\le\biggl(\sum a_n\biggr)^{1/2}\biggl(\sum\frac1{n^2}\biggr)^{1/2}.$ holds and squeeze theorem says that limits of left and right i'parts' should be equal. But here that may not be satisfied
If $X_1$, $X_2$, and $X_3$ are normal distributed with $N(5,2)$, and $X_4$, $X_5$, and $X_6$ are as well but with $N(4,4)$. Then, what is $P(X_1+X_2+X_3>X_4+X_5+X_6)$ if $X_1$, $\dots$, $X_6$ are independent?
"Without loss of generality we shall often assume that the homomorphism $v : F^\times \to \Bbb Z$ is surjective", the wlog just serves to remind the reader that it is always possible to normalise a discrete valuation right?
I first learnt group theory 1.5 years ago from A First Course in Abstract Algebra by John Fraleigh; then I learnt Galois theory 1 year ago from Ian Stewart; and then I learnt some commutative algebra 0.5 year ago from Atiyah-Macdonald; but of course the people here helped me a lot.
Actually I follow [this](https://math.stackexchange.com/a/2895315/579780)(read somewhere) as a definition of understanding, but, in abstract algebra all proofs seems to me as a combination of multiple ideas instead of a single idea. I don't know if I have fully understood it.
There is no amorphous set of reals - in fact, no amorphous set can be linearly ordered.
(For further discussion of what kind of structure amorphous sets, and more generally Dedekind-finite sets can have, see this paper of Truss or Agatha Walczak-Typke's Ph.D. thesis.)
To see this, suppose $A$ i...
