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"Nothing is more important than to see the sources of invention which, in my opinion, are more interesting than the inventions themselves."

--Gottfried von Leibniz (1646-1716)


7h
comment Uniform convergence of $\sum f(x)^n$
Have you seem the Weierstrass M-Test?
1d
comment Function differentiable on $(a,b)$ but not continuous on $ [a,b]$
$f(x)=1$ if $x\in(0,1)$, $f(0)=f(1)=0$.
1d
comment Show that a Series Diverges
Re: edit. No. You're thinking of $n$ as fixed downstairs in the first line, but not in the second.
1d
comment Integration by Substitution, can't solve (Working Added )
$x^3=(x^2+1-1)\cdot x$.
1d
comment If $\sum{a_k}$ converges, then $\lim ka_k=0$.
Useful observation: $n a_{2n}\le a_{n+1}+a_{n+2}+\cdots+a_{2n}$.
2d
comment Let $\mathcal H$ be a Hilbert space. If $\mathcal H$ is not finite-dimensional, then $B := \{x \in \mathcal H : ||x|| \le 1\}$ is not compact.
Compute the norm between two distinct members of your orthonormal set (or just the square of the norm).
2d
comment prove the limit of $k^{1/k}$ is $1$
See this.
2d
comment Give an example where $\{a^2_n\}_{n=1}^\infty$ is a Cauchy sequence, but $\{a_n\}_{n=1}^\infty$is not.
$1,-1,1,-1,\ldots$.
2d
comment Finite number of jump discontinuities
I'm imagining an infinite stair case with the width of the $m$'th stair being $1/2^m$, and each stair having height $1$.
Nov
24
comment Limit of $\frac{2^n}{3^{n+1}}$
$a_n=(1/3)(2/3)^n$.
Nov
24
awarded  Nice Answer
Nov
23
comment Check if set of functions is a basis of space
I don't think that's quite right. But, any linear combination of your functions has value zero, except for perhaps a finite number of points. The function $f(x)=x^3$ does not have that property.
Nov
22
comment Check if set of functions is a basis of space
Yes. But it's easy to do. Note any finite linear combination of your vectors gives a function that takes on only finitely many distinct values. (So, it's not a basis.)
Nov
22
comment Check if set of functions is a basis of space
Consider a function that takes on infinitely many distinct values.
Nov
22
comment How to check the compactness of these sets:
Each set contains the standard unit vectors. Does this set have a limit point?
Nov
22
comment Check if set of functions is a basis of space
Note you need to check if any given function is a finite linear combination of the $f_\alpha$...
Nov
22
comment How to prove that $\lim_{n\to\infty}{|\sin n|}$ doesn't exist
Hint: Each interval $[n\pi/2-1/2,n\pi/2+1/2]$ contains an integer.
Nov
21
comment Independence in Banach space
A trivial example: take $\ell_1(\Bbb R)$ and the standard unit vectors $\{e_\alpha:\alpha\in\Bbb R\}$. I'm not sure what happens in the separable case; but I'll think about it.
Nov
21
comment $\lim_{|x|\to\infty}f(x)=0$ implies $f$ attains its maximum value
First choose a closed ball, $B$, about $0$ so that $\Vert f\Vert$ is at most $\Vert f(0)\Vert$ on $B^C$.
Nov
20
comment How can I prove that a sequence such that every converging subsequence coverges to the same limit, converges?
For example the sequence $(x_n)$ with $x_n=n$ satisfies your hypothesis with $a=123833$.