Thomas E.
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 May2 answered Closure open ball May1 comment Show a bijection between sets What are your thoughts so far? Have you tried some candidates? Apr29 answered Prove that $S=\left \{ (x,y)\in \mathbb{R}^2:x>0\, \; or \; y>0\left. \right \} \right. \subset \mathbb{R}^2$is open Apr28 comment Is $A = f\mathbb{R^2}$ complete? $f:\mathbb{R^2}\to\mathbb{R^3}$, $f(x,y) = (x,y,x^2-y)$ @souf: Note that $f\mathbb{R}^{2}=\mathrm{graph}(g)$, where $g:\mathbb{R}^{2}\to \mathbb{R}$ as Goos defined above. So you only need to show that the graph of $g$ is a closed subset of $\mathbb{R}^{3}$ because closed subsets of complete spaces are complete. Apr28 answered Interior and closure of an arbitrary product Apr27 revised If $f$ is uniformly continuous on $\mathbb{R}$, $f(x) \ge a >0$ and $g(x) = 1/f(x)^2$, then $g(x)$ is uniformly continuous edited tags Apr13 comment Why do some series converge and others diverge? @Ethan: Because in the sum you add those numbers together. If you add numbers that are large then obviously the end result will be large. If you add negligent numbers together, then end result will stay small. Apr13 comment Why do some series converge and others diverge? @Ethan: I mean that the sum is always less than $1$, for any finitely many terms that you sum together. By converging fast enough to zero I mean the following. By looking at the sequences $(\frac{1}{n})_{n=1}^{\infty}$ and $(\frac{1}{2^{n}})_{n=1}^{\infty}$ you will see that both of them converge to zero, but e.g. the third term in the first sequence is $\frac{1}{3}$, whereas in the second sequence it is $\frac{1}{8}$: much less. And for any given $n\in\mathbb{N}$‚ the other sequence will always have small values than the other. The terms go faster to zero so to say. Apr13 comment Why do some series converge and others diverge? @Ethan: By a simple observation you can see that whenever you add finitely many of those terms $\frac{1}{2},\frac{1}{4},\frac{1}{8},...$ the sum is always bounded above by $1$. So the infinite sum must also be bounded above by $1$. The infinite sum is defined as the limit of the partial sums. they might be diverging or converging to a specific number, that all depends on the numbers that you're adding. Usually you can think that if the terms of the sum converge "fast enough" to zero, then the corresponding sum is finite. Apr11 revised Why do some series converge and others diverge? added 146 characters in body Apr11 revised Why do some series converge and others diverge? added 146 characters in body Apr11 answered Why do some series converge and others diverge? Apr10 comment Using Cantor's intersection theorem @Mitsos: It's false in general for continuous functions as well. It's true for example, always when $f$ is injective. You have to argue with what you know about the sets $F_{n}$ relationship to each other. Apr10 revised How to use Hölder's inequality to show $L_q$ is a subspace of $L_p$? deleted 21 characters in body Apr10 answered How to use Hölder's inequality to show $L_q$ is a subspace of $L_p$? Apr10 revised Homotopy on the unit circle added 190 characters in body Apr10 comment Using Cantor's intersection theorem Why can we conclude $\cap_{n}f(F_{n})=f(\cap_{n}F_{n})$? This is false in general. Apr10 answered Simply connectedness in $R^3$ with a spherical hole? Apr10 answered Homotopy on the unit circle Apr10 answered Is it possible to list $\mathbb{Q}$ so that the result set to be a monotonic sequence?