All Questions
6
questions
2
votes
3
answers
138
views
$|a-b|<\varepsilon\implies a=b$ for uniqueness of limit?
Consider the following statement:
Let $a,b\in\mathbb{R}$ be any two real numbers. Then we have that
$$\forall\varepsilon>0:|a-b|<\varepsilon\implies a=b.$$
Here is my attempt to prove it:
Proof. ...
- 77
2
votes
1
answer
134
views
Set of finite sums is dense
Let $(a_n)_{n\geq 1}$ be a sequence of non-negative real numbers such that $a_n \to 0$ but $\sum_{n\geq 1} a_n$ diverges. Show that the set of sums $\sum_{n \in S} a_n$, where $S$ ranges over the ...
- 2,733
1
vote
1
answer
75
views
$X_n - X_{n-2}\rightarrow 0$, prove that$ \frac{X_n}{n} \rightarrow 0$. [closed]
suppose $X_{n}$ is a sequence of real numbers such that $X_{n}-X_{n-2}\rightarrow 0$.
prove that $\frac{X_{n}}{n}\rightarrow 0$.
- 29
1
vote
2
answers
52
views
Show that $f(x):=\sum_{n=1}^{\infty}\frac{x}{(1+nx^{2})n^{\alpha}}\rightarrow 0$ as $x\rightarrow 0$, if $\alpha>\frac{1}{2}$.
Consider the function defined by $$f(x):=\sum_{n=1}^{\infty}\frac{x}{(1+nx^{2})n^{\alpha}}.$$
I have showed that, by Weierstrass M-test, when $\alpha>\frac{1}{2}$, the series converges uniformly to ...
- 5,930
6
votes
2
answers
1k
views
Defining $N$ in the $\epsilon$-$N$ definiton of convergence
In my Real Analysis class we've been spending some time talking about the $\epsilon$-$N$ definition of convergence. The book we are using, Elementary Analysis by Ross, defines convergence as:
A ...
- 331
1
vote
2
answers
102
views
Why the following sequence of function does not converge uniformly at $[0, \infty)$
Why the following sequence of function does not converge uniformly at $[0, \infty)$ but converge uniformly for some $a>0, [a,\infty)$
$$f_n(x) := n^2x^2e^{-nx}$$
So I know the limit function $f$ ...
- 2,167