0
votes
4answers
20 views

Why a sequence $\{x_k\}$ in $S$ such that $||x||>k$ cannot be a convergent subsequence?

This is an excerpt from my text: A set $S \subset \mathbb{R}^n$ is said to be compact if for all sequences of points $\{x_k\}$ such that $x_k\in S$ for each $k$, there exists a subsequence ...
0
votes
1answer
29 views

Sequence of functions that converges a.e. but not in the $L^1$ norm

How can I construct a sequence of functions that converges a.e. but it does not in the $L^1$ norm?
2
votes
1answer
32 views

The characteristic function induced by L^1 convergence function

Assume $f_n\in L^1(\Omega)$, $f\in L^1(\Omega)$, and $f_n\to f$ in $L^1(\Omega)$ where $\Omega\subset R^N$ is open. Define $$ E_t^n:=\{x\in\Omega, f_n(x)>t\}.$$ Hence we have \begin{equation} ...
4
votes
2answers
74 views

Prove that the sequence $\left\{\frac{n}{n^2+1}\right\}_1^\infty$ converges to 0.

Prove that the sequence $\left\{\frac{n}{n^2+1}\right\}_1^\infty$ converges to 0. I'm missing something. I doubt that I'm really doing anything from " $\frac{1}{n+1}$ " on. I set it $< ...
2
votes
1answer
27 views

Convergence of alternating series not subject to alternating series test

Given two $a_n,\; b_n>0$ such that $lim_{n \to \infty} a_n,\; b_n=0$ and $\lim_{n \to \infty} \frac{a_n}{b_n}=1$, where neither series is necessarily monotonic: if $\displaystyle \sum_{k=1}^\infty ...
2
votes
5answers
96 views

Determining convergence or divergence of series

I am wondering the convergence or divergence following series $$ \sum_{n=1}^{\infty} \frac{(-1)^n}{ n+\sin (n)} \\ $$ My 1st attempt is 'alternating series test' $$ $$ But, ...
1
vote
2answers
51 views

Taylor series of Infinitely differentiable function with nonnegative derivatives

Let $f(x)$ be a nonnegative and infinitely differentiable function on $[-a,a]$ to $\mathbb{R}$ such that $\forall x\in[-a,a]:f^{(n)}(x)\ge0$. Prove that the series: $$\sum_{i=1}^\infty ...
2
votes
2answers
28 views

Suppose that $f(x) \ge 0$ and $\lim_{x \to c} f(x) = L$. Prove $\lim_{x \to c} \sqrt{f(x)} = \sqrt{L}$

Suppose that $f(x) \ge 0$ in some deleted neighborhood of $c$, and that $\lim_{x \to c} f(x) = L$. Prove that $\lim_{x \to c} \sqrt{f(x)} = \sqrt{L}$ under the two different assumptions on $L$: $L=0$ ...
0
votes
1answer
27 views

Cauchy sum of ratio of sequences not converging.

Prove if ${x_n}$ and ${y_n}$ are Cauchy and $x_n + y_n > 0$, for all natural n, then $\frac{1}{x_n + y_n}$ cannot converge to zero. Attempt: Suppose $x_n → a$ and $y_n → b$. Then $x_n + y_n → a + ...
1
vote
1answer
12 views

Bounded intervals, sequentially compact.

Can someone please give me an example of a bounded interval in R that are not sequentially compact? A subset E of R is said to be sequentially compact if and only if every sequence x_n in E has a ...
3
votes
1answer
52 views

Proving convergence of $\sum \frac{\sin n}{2^n}$

Prove that the following sequence ($x_n$) is convergent: $$ x_n = \frac{\sin 1}{2} + \frac{\sin 2}{2^2} + \frac{\sin 3}{2^3} + ... + \frac{\sin n}{2^n} $$ I have tried to use to the sequence is ...
1
vote
2answers
24 views

$f_n$ converges uniformly on $\overline\Omega$

Suppose $\Omega$ be a bounded region and $\{f_n\}_{n\in\mathbb N}$ a sequence of continuous functions on $\overline\Omega$ which are holomorphic in $\Omega$ and $f_n$ converges uniformly on the ...
1
vote
1answer
56 views

Prove that x{n} is convergent

So I'm currently studying for my midterms and I found the following question from my practice set which I'm unable to solve: Prove that the following sequence $(x_{n})$ is convergent. Let $$x_{n} = ...
0
votes
2answers
47 views

Prove that limit exists and is finite, sum of series, Cauchy.

Prove that $\lim_{n \to \infty} \sum_{k=1}^n \frac{(-1)^k}{k}$ exists and is finite. Attempt: Suppose $\{x_n\}$ is real sequence, and $x_n = \frac{(-1)^k}{k}$. I know if I prove that it is Cauchy, ...
2
votes
1answer
47 views

Simple proof that this sequence converges [verification]

This is a relatively simple problem. I'm just making sure I have the right idea here. I'd like to prove that the sequence $\displaystyle a_n = 1 + \frac{1}{n^{1/3}}$ converges. My proof is: We ...
5
votes
2answers
126 views

Picard Iterates Converge Uniformly

I have a homework question that asks to show that the Picard iterates $$ \phi_{n+1}(t) = \int_0^t 1 + \phi_n^2(s) \, ds, \quad \quad \phi_0(t) = 0 $$ converge uniformly on any compact interval $[-r, ...
0
votes
2answers
23 views

What does this function converge to in $\mathbb{R}$ equipped with discrete metric?

We're given this function $f_n (x) = \begin{cases} 0 \ \mbox{ if $x <1/n$}\\ 1 \ \mbox{ if $x \geq 1/n$} \end{cases}$ I think it converges pointwise to $f(x) = \begin{cases} 0 \ \mbox{ if $x ...
0
votes
1answer
42 views

Visual understanding of convergence of domains in the sense of Fisher

In these lecture notes by Ueltschi here, I found in Definition 2.3 a peculiar type of convergence. Especially the second property is hard for me to visualize what it means, could anybody try to ...
2
votes
1answer
32 views

Pointwise convergence implies uniform convergence in discrete metric space $(X,d)$?

In general, does this hold for a sequence of functions in an arbitrary $X$? For a sequence to converge in the discrete metric, the sequence needs to become a constant sequence for a sufficiently large ...
-1
votes
1answer
28 views

Proving that a sequence is Cauchy on the basis of squeeze theorem

Let $\{x_n\}$ be a sequence of real numbers such that $$|x_n| \leq \frac{2n^2 + 3}{n^3 + 5n^2 +3n + 1}$$ Prove $\{x_n\}$ is a Cauchy sequence Proof: Suppose that ${x_n}$ be a sequence of real ...
1
vote
1answer
39 views

Is there a contradiction in these two definitions of limit Superior?

Definition 1 : Let $A=\{a_n\}$ be a sequence of real numbers not necessarily bounded . Then we define : $\lim \sup ~ a_n = \inf ~\{\sup ~a_n,~\sup ~a_{n+1}~\sup ~a_{n+2}, \cdots\} $ Definition 2 : ...
4
votes
1answer
89 views

Cauchy Sequences, not converging to zero

True or False? If $\{x_n\}$ and $\{y_n\}$ are Cauchy and $x_n + y_n > 0$, for all $n\in\mathbb{N}$, then $\left\{\frac{1}{(x_n + y_n)}\right\}$ cannot converge to zero. I believe the claim to be ...
0
votes
1answer
42 views

alternating series $\sum(-1)^na_n$ is divergent, then, is $\sum A_k$ divergent?

An alternating series $\sum\limits_{n=1}^\infty (-1)^na_n$ is divergent , $a_n\geq0$, and $\lim\limits_{n\to\infty}a_n=0$. Could we conclude that $\sum\limits_{k=1}^\infty A_k$ is divergent, too ? ...
3
votes
1answer
47 views

Proving Archimedes Sequences Equal $\pi$.

I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge ...
0
votes
1answer
37 views

Proving a sequence is increasing and converging as $n\to \infty$.

Suppose that $x_0 \in (-1,0)$ and $x_n=\sqrt{x_{n-1}+1}-1$ for $n \in \mathbb N$. Prove that $x_n \uparrow 0$ as $n\to \infty$. What happens when $x_0 \in [-1,0]$? Before this, the problems I did had ...
2
votes
1answer
39 views

Sequence, Monotone Convergence Theorem.

Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1} - 2}$ for all natural $n$. Use the Monotone Convergence Theorem to prove that either $x_n \rightarrow 2$ or $x_n \rightarrow 3$ as $n$ grows. ...
1
vote
1answer
27 views

Decreasing sequence, bounded below.

Suppose that $0 \leq x_1 < 1$ and $x_{n+1} = 1 - \sqrt{1 - x_n}$ for all natural $n$. Prove that $x_n$ is decreasing and bounded below as $n$ converges. Attempt: Suppose that $0 \leq x_1 < 1$ ...
1
vote
1answer
31 views

Does the sequence converge or diverge?

I'm having trouble understanding how to do this one. If anyone could help I would be grateful. Does the sequence $$ \left\{ \sum_{n=1}^k \left( \frac{1}{\sqrt{k^2+n}} \right) ...
3
votes
0answers
67 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
0
votes
0answers
54 views

Convergence of the series $\sum_{n=1}^\infty (1+\frac{1}{\sqrt{n}})^{-n^\frac{3}{2}}$

Test the convergency of the series $$\sum_{n=1}^\infty \left(1+\frac{1}{\sqrt{n}}\right)^{-n^\frac{3}{2}}.$$ We know that, if $\sum_{n=1}^\infty U_n$ is convergent, then ...
0
votes
3answers
48 views

If $X=\{x_n\}, Y = \{y_n\}$ be bounded sequences of real numbers. Then, if $x_n \leq y_n~\forall~n$, then $\lim \inf (x_n) \leq \lim \inf(y_n)$

If $X=\{x_n\}, Y = \{y_n\}$ be bounded sequences of real numbers. Then, if $x_n \leq y_n~\forall~n$, then show that $\lim \inf (x_n) \leq \lim \inf(y_n)$ and $\lim \sup (x_n) \leq \lim \sup (y_n)$ ...
1
vote
0answers
21 views

If in an upper bounded real sequence $X : v_m =\sup \{x_n : n \geq m\}$, then $x^* = \lim \sup X = \inf \{v_m : m \geq 1\}$

If in a bounded real sequence $X : v_m =\sup \{x_n : n \geq m\}$, then : $x^* = \lim \sup X = \inf \{v_m : m \geq 1\}$ Proof Attempt : If $X= \{x_n\}$ is a sequence of real numbers which ...
0
votes
1answer
11 views

Not existance of one sided limit

I have a question regarding one sided limits of functions. Let's say that the function $f$ is defined in $(a,b)$. And let's say that we want to check the limit of $f$ when it approaches b from the ...
3
votes
2answers
68 views

prove a sequence without knowing its convergence

Let $(X_n)$ be the sequence with $X_1=2$ and $X_n=\sqrt{5X_{n-1} + 6}$ for all $n\ge 2$. How can you prove that it is convergent? IF given convergence, I know that its limit is $6$, but the question ...
0
votes
0answers
16 views

Prove that if $\{a_n\}$ converges, then $\{a_{2n+1}\}$ converges, using definition of convergence [duplicate]

Use the definition of convergence to prove that if the sequence $\{a_n\}$ converges, and $b_n = a_{2n+1}$, then the sequence $\{b_n\}$ also converges. Progress I was thinking about using proof by ...
0
votes
1answer
23 views

Comparing the order of convergence $\mathcal{O}( h^2 |\log(h)|)$

I don't have any intuition in judging how fast a term of the order $\mathcal{O}( h^2 |\log(h)|)$ is decreasing as $h \to 0$, so i tried comparing it with terms of the form $\mathcal{O}( h^\alpha )$ ...
3
votes
0answers
82 views

Convergence in probability and convergence of Cesaro means

Consider the random variable $X_n$, not necessarily iid. If $X_n\rightarrow 0$ almost surely, then the Cesaro means $\frac 1n\sum_{k=1}^nX_n$ converge almost surely to 0. This cannot be weakened to ...
0
votes
1answer
26 views

Distance between two compact subsets is always $\geq$ distance between two particular points of the subsets

Show that if $K_1$ and $K_2$ are compact subsets of $\mathbb R^p$, then there exist points $x_1$ in $K_1$ and $x_2$ in $K_2$ such that if $z_1$ belongs to $K_1$ and $z_2$ belongs to $K_2$, then ...
3
votes
1answer
31 views

Help Understanding Step in Proof of Convergence

The theorem is If $\sum a_n$ is a series of complex numbers which converges absolutely then every rearrangement of $\sum a_n$ converges, and they all converge to the same value. The proof ...
0
votes
1answer
37 views

what is the difference between bounded and convergent?

I know that bounded means to have an upper or lower bound. Let $E \subset \mathbb{R}$ be nonempty. The set $E$ is said to be bounded above if and only if there is an $M \in \mathbb{R}$ ...
1
vote
0answers
15 views

scalar dimension to the approximation of an integrable function

Let $(M,\mathcal{A},\mu)$ space of probability. If $f\in L^1(\mu)$ then $f=\displaystyle{\lim_{n \rightarrow +\infty}\sum_{i=1}^n}\alpha_i\chi_{C_i}$ where $C_i\in \mathcal{A}$ and $\alpha_i \in ...
0
votes
1answer
27 views

Assumption in a convergence proof

Im in the middle of a proof of the fact that for a>0, if lim $x_n$ = a, then lim $\sqrt{x_n}$ = $\sqrt{a}$. I'm in the step that i use $| \sqrt{x_n} - \sqrt{a} |$ = $ \frac{|x_n - a|}{ ...
0
votes
2answers
43 views

If $x = lim (x_n)$ and if $|x_n - c| < \epsilon~~ \forall ~n \in N$, then it is true that $|x-c|< \epsilon$

If $x = lim (x_n)$ and if $|x_n - c| < \epsilon~~ \forall ~n \in N$, then is it true that $|x-c|< \epsilon?$ I am a little confused about this question ( it appears in the Bartle's elements of ...
0
votes
1answer
55 views

Find a Cauchy sequence that does not converge

I am supposed to look at $l_0$, the set of all sequences with finitely many non-real elements in $(l_0,d_{\infty})$. It is just that I don't quite understand how the $d_\infty$-metric is defined on ...
2
votes
2answers
17 views

approximate a Borel set by a continuous

I wonder if it is possible to approximate a Borel set by a continuous function i.e. Let $B$ a Borel set in $(X,d)$ (compact separable metric space) I wonder if there continuous functions ...
0
votes
1answer
37 views

Why does these complex sequences converge uniformly?

I have one complex series and one sequence. It is used in complex analysis in a part of my book where they are integrated. However, as you know in order to change limit and integration order it has to ...
1
vote
0answers
31 views

solution of the set of real non-linear equations

I have a set of real non-linear equations as following: \begin{equation} y_0 = f(y_0,y_1) \\ y_1 = g(y_0,y_1,y_2) \\ y_2 = g(y_1,y_2,y_3) \\ \vdots \\ y_{n-1} = g(y_{n-2},y_{n-1},y_n) \\ y_n = ...
0
votes
2answers
30 views

A query in the proof of convergence of the set $\{1/n\}$

I have a query regarding the proof of the statement that the set $S = \left\{ \dfrac {1}{n} \right\}$ has limit point $0$. I am studying an introductory course in Analysis. Proof: From the ...
2
votes
1answer
44 views

Find the limit of $\frac{1}{n^2}\sum_{k=1}^n k^2\sin\left(\frac{\theta}{k}\right)$

I am trying to find the limit of $$\frac{1}{n^2}\sum_{k=1}^n k^2\sin\left(\frac{\theta}{k}\right)\qquad \theta>0$$ as $n\to \infty$. I know that since $x-\frac{x^3}{6}\leq \sin x \leq x$ then ...
0
votes
1answer
32 views

Accumulation points of trigonometric sequences

I am interested if the following sequences have accumulation points: $$x_{n} = \sin(2+\frac{1}{n})$$ and $$x_n = \tan (n)$$ Specifically for the first sequence is $1$ and $-1$ accumulation points? ...