aye, there's between 8 and 20mg of nicotine in a cigarette and I usually have 6mg in a full tank of liquid and never experienced any craving for cigarettes
thankfully I haven't developed any form of heart issues (even tho I had been smoking at that level for about 5 years), and that's probably due to the amount of cardio I do, but my lungs scream if I stop cycling for like a week
Probably an elementary topology question here: let's suppose I have a separable metric space $X$, and let $(x_n)_{n=1}^{\infty}$ be a dense sequence in $X$. My professor writes: $\forall \epsilon > 0, X = \bigcup_{n=1}^{\infty}B(x_n, \epsilon)$. Where can I find a reference for this result?
But if you're going to take measure theory to take graduate level probability, you need to fill in the prerequisites. You and all the rest of the horde.
So Munkres says here that $A \subset X$ is dense if $\bar{A} = X$. So if I'm understanding that, $\bar{A}$ is the closure of $A$. Now I gotta look that one up...
Theorem 17.6 in Munkres, thank you @Ted. So the closure of $\bar{A}$ is equal to the union of $A$ itself and the set of its limit points.
Thus, in the case of these dense subsets of $X$ denoted $(x_n)_{n=1}^{\infty}$, we know that their closures must be equal to the unions of the $x_n$s themselves as well as their corresponding limit points, resulting in those open balls.
Yeah, if you have a point that's at least $\epsilon$ away from the $x_n$, it can't be in the closure of the $\{x_n\}$, so what you have as a result are open balls of length $\epsilon$ around each $x_n$
Directly, @Clarinet, if you take any ball of radius $\epsilon$ about an arbitrary point $x$, it must contain some $x_n$ (that's what being a limit point tells you). But this means, then, that $x$ is in $B(x_n,\epsilon)$.
Here's what I'm going to do - I'm going to finish scribing this lecture and probably just spend this week actually doing quality work on topology. Miraculously, I managed to finish my HW a week early (will have to check for typos soon), but it's really clear I'm going to have to get some of us together to spend a few nights reviewing topology. I need to understand these terms better and it can't wait much longer.
It's also really clear that I'm going to need to know much more than the basics.
I really wish I could have taken topology in my undergrad. It was only offered once every two years at my alma mater.
@TedShifrin I'm aware of that. I did some reading on lim infs and lim sups this last summer. Perhaps I'm wrong, but I suspect the prof I have has opted to skip that material
It's really odd to me how diverse measure theory w/probability books are with the intro material
Anyway, I've definitely had an appreciation for how tough this subject is now. It seems like you'd have to really know what you're doing if you're teaching this class given how different approaches can look. I can appreciate how difficult that is.
I honestly thought I was just incompetent given the fact that I have... what, 20 measure theory books in my shelf, have tried to read them, and often couldn't get past the first chapter
If you have $(X, \lambda)$ a probability measured space and you look at all a.c. probability measures on this space that's exactly the same as continuous functionals on $X$ which integrate to $1$
let's say we are dealing with a proof by induction, then show that P(0) holds, then in the inductive hypothesis assume that P(i) holds for i = {0, ..., n} for some n and then in the inductive step we use an argument that if we would plug n = 0 (i.e, P(0)) then we get something that is not in the inductive hypothesis (i.e., P(-1))
would this be considered to be a flaw in the proof, or since we have the base P(0) and we showed it holds, it will be considered valid?
Quick question: is this just floating-point error, exp(x) approximation error, cos(x) approximation error, some combination of these, or what? desmos.com/calculator/9wvgqabq3y
I decided to try and just work on improving my existing approximation.
Ok I'm convinced that's probably just FP error down at 10^-15 there
@JoeShmo If we are trying to prove for example that $T(k)$ represents the equality $i^k = 1, i \neq 0$. Would the following proof fail for $i = 0$ ($T(0)$)? Here is the proof: BS: $T(0): i^0=1$ is certainly true. IH: Suppose $0 \leq j < k$. IC: Note that $i^k \frac{i^{k-1}i^{k-1}}{i^{k-2}}$ by exponents rules, thus, $k = k-1+k-1-(k-2)$. Now, turn you attention to $k-1 < k$ and $k-2 < k$ so we can apply our IH and get $i^{k-1}=i^{k-2} \Rightarrow i^k = 1$
Anyway, I really wish this measure theory prof would just keep to one textbook. As I type this, I have about 15 textbooks out, mix of measure theory and topology books
