38m
comment A limit of a sequence
Here's one duplicate.
13h
comment Prove that there exists $c \in \mathbb{R}$ such that $f'(c)=0$ if $ \lim_{x\to-\infty}f(x) =\lim_{x\to\infty}f(x) = 0 $.?
Use Rolle's Theorem.
15h
comment A Challenge on linear functional and bounding property
@hardmath Presumably, "bounded" means "continuous" in the topological sense.
1d
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 $x\in(0,1)$, $f(0)=f(1)=0$.
2d
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.
2d
comment Integration by Substitution, can't solve (Working Added )
$x^3=(x^2+1-1)\cdot x$.
2d
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).
Nov
25
comment prove the limit of $k^{1/k}$ is $1$
See this.
Nov
24
comment Give an example where $\{a^2_n\}_{n=1}^\infty$ is a Cauchy sequence, but $\{a_n\}_{n=1}^\infty$is not.
$1,-1,1,-1,\ldots$.
Nov
24
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.
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