# Tagged Questions

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### Limit of a sequence proof by contradiction

Suppose I have a monotonically decreasing sequence $a_{n}$ such that $a_{n}$ is positive for all $n \in \mathbb{N}^{+}$ and that $$\lim_{n\rightarrow \infty} \frac{a_{n+1}}{a_{n}} = 0.$$ It seems to ...
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### If $a_n=n^\frac 1n-1, n \in \mathbb N$ prove that $0 \le a_n \le \sqrt {2/n}$?

If $a_n=n^{\frac{1}{n}}-1$, $n\in\mathbb{N}$, prove that $0\le a_n\le\sqrt{\frac{2}{n}}$. I tried with induction and signs, got nowhere. Any help is appreciated.
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### Show that $\lim_{n\to\infty}\frac{a^n}{n!}=0$ and that $\sqrt[n]{n!}$ diverges.

Let $a\in\mathbb{R}$. Show that $$\lim_{n\to\infty}\frac{a^n}{n!}=0.$$ Then use this result to prove that $(b_n)_{n\in\mathbb{N}}$ with $$b_n:=\sqrt[n]{n!}$$ diverges. ...
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### Proving a property of the largest limit point

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence. By Bolzano-Weierstraß this sequence does have a limit point. Let $\bar{a}$ denote the largest limit point of the sequence. Show that among ...
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### Proving $\{x_n\}$ converges to $a$ when $|x_n-a|\le Cb_n$ for large $n$ and $C$ is a positive constant.

If $\{b_n\}$ is a sequence of nonnegative numbers that converges to $0$, and $\{x_n\}$ is a real sequence that satisfies $|x_n-a|\le Cb_n$ for large $n$, where $C$ is a fixed positive constant, prove ...
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### Prove $x_n$ converges to $a$ iff every subsequence of $x_n$ also converges to $a$.

Prove $x_n$ converges to $a$ iff every subsequence of $x_n$ also converges to $a$. Suppose that $\{x_n\}$ is a sequence in $\mathbb R$. Definitions available: (1) A sequence of real numbers ...
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### Attempt to prove that every real number is a limit of a sequence of rational numbers

Prove that given a real number $x$, there exists a rational sequence $r_n$ such that $r_n \to x$ as $n$ grows. Proof: Suppose $x$ is a real number. Then we know by definition, there exists a ...
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### Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation $$\sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2}$$ defined for $0\le x\le 2\pi$. Here is ...
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### $(u_{2n})$,$(u_{2n+1})$,$(u_{3n+1})$ converge $\underset{???}{\Rightarrow}(u_n)$ converges

Let $(u_n)_{n_\in\mathbb{N}}\in\mathbb{C}^\mathbb{N}$. We know that $(u_{2n})$, $(u_{2n+1})$ and $(u_{3n+1})$ converge. The question is to know whether $(u_n)_{n\in\mathbb{N}}$ converges. /!\ I ...
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### Convergence of a sequence in absolut value.

I need to prove this: If $a_{n}$ converges to $A$, then $|a_{n}|$ converges to $|A|$. And I have this: $a_{n} \rightarrow A$ then, given $\epsilon>0$ there exists $N \in J$ such that ...
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### A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit

In one of my books there was an exercise to prove that: A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit (I know a sequence doesn't really converge to ...
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### Sequential compactness in $\mathbb{R}$

Well known result: Suppose $f:\mathbb{R}\to \mathbb{R}$ is continuous and let $K$ be a compact set. Then, $f(K)$ is compact. I can prove this using the definition of compactness (finding a ...
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### If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N.$$ Suppose there exists a continuous function $g$ on ...
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### For what $\alpha$ does the series converge: $\sum^\infty_{n=2}\frac {1}{n^\alpha\log_2(n)}$

Let $\alpha\ge 0$ check for what $\alpha$ does the series converge: $$\displaystyle\sum^\infty_{n=2}\dfrac {1}{n^\alpha\log_2(n)}$$ I tried the condensation test and get: ...
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### $a_{n+1}=a_n-a^2_n$ show the recursion sequence is convergent and find its limit

Let $a_1=\frac 2 3 , \ a_{n+1}=a_n-a^2_n$ for $n\ge 1$. Show the sequence is convergent and find its limit. In order to show convergence, I need to show that it's monotone and bounded. ...
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### Proof Check: Every Cauchy Sequence is Bounded

Sorry if I keep asking for proof checks. I'll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I'd do a proof that made sense ...
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### Prove that any unbounded sequence has a subsequence that diverges to $∞$.

To prove that any unbounded sequence has a subsequence that diverges to ∞, is it enough to say that you can take a subsequence $(a_{m(k)})$ where $m(k)=k$, as you know that this diverges to infinity, ...
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### I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
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### Less than or equal summations

Hi,I want to prove above unequal that consist of two summation both of this sides.It a formula in Computer Network to control Congestion.The way to prove it is not important, but because I weak in ...
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### Is any of this true about infinite series of functions?

Let $f_n^+(x)$ be a sequence of non-negative functions $f_n^+: X \to \Bbb{R}_{\geq 0}$, such that each $f_n^+$ has countably many zeros. Then if $f(x) = \sum f_n^+(x)$ converges point-wise, the ...
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### Find maximal possible sum of a tricky series

for $a \in R$, $n \in N$ let $a_n$ closest distance between $a$ and $\frac m {2^n}$, where $m \in Z$. Find maximal possible sum of a series: $\sum_{n=0}^\infty a_n$ I came up with solution for the ...
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### Non-increasing Monotone Sequence Convergence Proof

My goal is to prove the monotone convergence of a non-increasing sequence of real numbers. There are some steps in the proof that I'm not sure about. The question: If $S$ is a non-increasing sequence ...
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### Proof that if $\lim s_n=0$ then $\lim\sqrt{s_n}=0$

Here's the question: Suppose $(s_n)$ is a sequence of non-negative real numbers, and $\lim( s_n) = 0$. Prove that $\lim(\sqrt{s_n})=0$. Here's my proof. Can someone please verify it or offer ...
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### If a sequence of functions converges uniformly, then its limit is bounded

Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of functions defined on $[a,b]$ Assume that each $f_n$ is bounded, so $|f_n| \le M_n$ for all $x \in [a,b]$. If $\{f_n\}_{n=1}^{\infty}$ converges uniformly ...
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### Proving $\sum^\infty_{n=1}a_n$ converges absolutely iff each each sub series converges

We have a series $\displaystyle\sum^\infty_{n=1}a_n$ and a sub series $\displaystyle\sum^\infty_{k=1}a_{n_k}$ where $n_k\in\mathbb N$. Prove that $\displaystyle\sum^\infty_{n=1}a_n$ converges ...
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### Convergence of geometric shapes

Let $c$ be a closed geometric shape. Let $P$ and $Q$ be two points in $c$. Let $S$ be the family of all closed squares. Let $s_1, s_2,...$ be a sequence of shapes from family $S$ such that for every ...