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Apr
22
comment If $a_n = \frac{e^{n}}{e^{2n}-1}$ how do I show that $a_{n+1} \leq a_n$?
@EasyStarter using $\sinh$ is about as direct a method as you could hope for.
Apr
12
comment Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$
@RobertLewis thanks : )
Apr
8
comment Is a compact, simply-connected 3-manifold necessarily $S^3$ with $B^3$'s removed?
Modulo this "half lives, half dies"-business, you've convinced me about the fact that the boundary is composed of spheres.
Apr
6
comment $\frac{1}{2}!$ aka $\Gamma(\frac{3}{2})$
$\Gamma(\frac12)=\sqrt{\pi}$ is a classical result, and the fuctional equation $x\Gamma(x)=\Gamma(x+1)$ gives you the result. Is this the sort of thing you are looking for?
Apr
5
comment Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$
Have you looked at some standard textbooks?
Apr
5
comment Continuity of left derivative implies differentiability?
The question you link to is very related, but it's not quite the same, as only continuity at one point $x$ is hypothesized. Can the solution be modified?
Apr
4
comment Show that no non-trivial open set in $\mathbb{R}^n$ has measure zero in $\mathbb{R}^n$
@user228695 $A$ isn't open in $\Bbb R^2$.
Apr
2
comment $\sum a_{2n} $ converges
Useful examples of converging, non absolutely converging series arise from alternating series. I wonder why people are downvoting...
Apr
2
comment Is a variety a CW-complex?
For manifolds this is part of Morse theory.
Apr
1
comment Differential-Geometry question- Curve Theory
No problem : ) ${}$
Apr
1
comment Differential-Geometry question- Curve Theory
Then why not edit the question accordingly?
Apr
1
comment Differential-Geometry question- Curve Theory
Should this be an equality?
Mar
31
comment Let $|f(z)| \to \infty$ as $|z| \to \infty$, prove that $f(\mathbb{C})= \mathbb{C}$?
@Socchi for any $a\in\Bbb C$ and $\rho>0$ set $\overline{D(a,\rho)}=\lbrace z\in\Bbb C\mid|z-a|\leq\rho\rbrace$The $r$ comes from the fact that $\overline{D(0,R)}$ is compact, so is its image under the continuous map $f$, and so it is closed, i.e. its complement is open, and $a$ lies in that complement.
Mar
31
comment Let $|f(z)| \to \infty$ as $|z| \to \infty$, prove that $f(\mathbb{C})= \mathbb{C}$?
@Chappers Continuity gives you this.
Mar
31
comment Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$
@user123733 Draw a picture of a sequence $k_n$, hopefully this will make things clearer.
Mar
31
comment Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$
Just take $\epsilon=\frac{b-a}2$. The reasoning above shows that there exists some $i$ with $k_n\in(a,b)$.
Mar
31
comment Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$
@user123733 The key point is that the $1-\epsilon\leq k_0\leq 1$, and for all $i$, $0<k_{i}-k_{i+1}\leq \epsilon$ and $k_i\to 0$. This means that any $c$ is $\epsilon$-close to some $\phi(n)/n$
Mar
30
comment Example of $H^n(X,R)$ not equal to $Hom(H_n(X,R),R)$
@Dan You seem to suggest that my question was designed merely as a way to show off, when in reality I was trying to rule out either a confusion or a typo: the universal coefficient theorem, at least the one I know, relates homology with coefficients in a PID to cohomology with coefficients in a module over said PID. How is suggesting that there may be a $\Bbb Z$ (or any other PID for that matter) rather than a $G$ for the coefficients of homology a rude question when the question doesn't seem to make sense without it?
Mar
30
comment Example of $H^n(X,R)$ not equal to $Hom(H_n(X,R),R)$
Why are there two $G$'s in the $Hom$ both in the title and the body of the question)? When you replace the first $G$ with $\Bbb Z$ in the title, then this is an isomorphism, because $H_0$ is free abelian group, and thus the $Ext^1$ vanishes.
Mar
27
comment Vector bundle of dimension $\leqslant n$ on $n$-connected space is trivial
You can have an $n$-skeleton equal to a point, not just the $(n-1)$-skeleton.