1
vote
2answers
26 views

A Question on Bolzano-Weierstrass Theorem

The following are two equivalent statements about the Bolzano-Weierstrass theorem. Every bounded real sequence has a convergent subsequence. A subset of $\mathbb R$ is compact if and only if it is ...
0
votes
1answer
20 views

Convergence of $\sum \frac{i^n}{\log(n)}$

Study the convergence of $$\sum_{n \geq 2 } \frac{i^n}{\log(n)} $$ where $\log$ of course denotes the 'natural logarithm' and $i \in \mathbb{C}$ Oddly enough I managed to show Abel's Test for ...
-1
votes
1answer
26 views

How to derive this inequality

I learnt that for a standard normal random varialbe $Z$ and positive $x$, we have $$\mathbb P (Z > x) \geq \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{1}{x+\frac{1}{x}} ...
2
votes
0answers
66 views

measure theory exercise (verification)

Hi I found the following exercise in the Dudley's book and I'd like to see if my answer is correct; the last part is what I'm not entirely sure, since I'm not completely familiar with this kind of ...
0
votes
1answer
10 views

on Limits and sequences proofs of a closed subspace of the hilbert space.

$M$ is a closed subspace of the Hilbert space $H$ and $ x\in H$ My book states these two claims. (1) If $d = \inf_{y \in M} \|x - y \|^2 $then there is a sequence of elements $\{y_n\}$ of $M$ such ...
6
votes
0answers
115 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
1
vote
1answer
30 views

$\{ x \in H: x=\sum_{k=1}^{\infty}c_{k}u_{k}$, $|c_{k}| \leq \frac{1}{k}\}$ is compact

Let $H$ be a complex inner product space that is also a complete metric space with respect to the distance induced by the inner product. Assume $\{u_{k}\}_{k=1}^{\infty}$ be an orthonormal set in ...
2
votes
1answer
51 views

A question on Lebesgue measure: Inequalities

Let $A_n$ be a decreasing sequence of Borel sets with finite but with measure $\geq \epsilon > 0$ for all $n$. Then there exists a compact set $B_n \subset A_n$ for all $n$ such that $\mu(A_n ...
2
votes
1answer
65 views

Arzela Ascoli, help to understand some points in the proof.

Hi everyone I'd like if someone could give me an explanation of some points in the following proof, explicitly the points with the asterisk. This is from Dudley's, one direction is completely easy, ...
1
vote
1answer
16 views

Expectation of Truncated Random Variables (2)

This is a follow-up question from here. Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $$0<\delta<0.5$$ ...
5
votes
3answers
67 views

About using Archimedean property to prove the existence of least upper bound

I am studying the proof of existence of least upper bound, but I can not understand how the autor applies the Archimedean property. Archimedean property. Let $x$ and $\epsilon$ be any positive ...
0
votes
1answer
31 views

Follow-up question on mathematical induction with arbitrary base case

Note: This question has already been answered here Proving mathematical induction with arbitrary base using (weak) induction. I was trying to 'reconstruct' at least one proof given in this question ...
4
votes
4answers
62 views

Existence of the square root in $\mathbb{C}$

I am stuck on the following Proposition: Proposition: Show that for every $z \in \mathbb{C} \setminus (- \infty, 0]$ there exists exactly one $w \in \mathbb{C}$ such that $w^2=z$ and Re$(w)>0$ ...
0
votes
1answer
21 views

On the equivalency of two indefinite integrals using u substitution.

I am reading the Separation of variables page on wikipedia, at a certain point it states that the following equation Is equal to (1) because of the substitution rule of integrals. The ...
4
votes
0answers
63 views

How to select good exercises?

I'm studying on Rudin "Principles of of Mathematical Analysis" which I begin to find as a good and complete reference. I wonder how many exercises shall I do at the end of each chapter ? In case of ...
1
vote
1answer
41 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
2
votes
4answers
100 views

Constructive proof for existence of integer part of real number

I try to prove de following exercise of my analysis textbook. Show that for every real number $x$ there is exactly one integer $N$ such that $N \le x < N + 1$. I have been finding a ...
0
votes
1answer
31 views

Definition of $C^1$ functions with values in $\mathbb R^m$

My analysis textbook defines a $C^1$ function $f:\mathbb{R^n}\to\mathbb{R^m}$ as one in which for each component function $f_i, 1\leq i\leq m$ the partial derivative $\frac{\partial f_i}{x_j}$ exists ...
2
votes
1answer
26 views

Fourier Transforms of $L^1$ functions

Suppose that $f_n$ and $f$ are $L^1(\mathbb R^n)$ functions with $f_n \to f$ in $L^1$ sense. Then is it true that their Fourier transforms defined as $$ \hat f(\xi) := \int_{\mathbb R^n} ...
3
votes
0answers
73 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 ...
1
vote
1answer
33 views

Convergent Series $\frac{1}{n^q}, \ \ q>1$

How to show the following result about series? Thank you! Convergent Series: $$\sum_n \frac{1}{n^q}, \ \ q>1$$
0
votes
3answers
40 views

Finding a recursive formula for a number

I am trying to find a recursive formula for a given number in order to solve a problem I am working on. For every $n \in \mathbb{N} \setminus \lbrace 0,1 \rbrace$ we define the number ...
0
votes
0answers
25 views

On a step in a solution of a proof of the Continuity of thomae's function at irrationals.

Thomae's function: $$t(x) = \begin{cases} 0 \ \ \text{if} \ x \in \mathbb{R}- \mathbb{Q} \newline \frac{1}{q} \ \ \text{if} \ x \in \mathbb{Q} \ \text{and} \ x = \frac{p}{q} \ \text{in lowest terms} ...
0
votes
2answers
34 views

Proof of: Because $I$ is dense in $R$ there exists a sequence $\{x_n\} \subset I$ with $\{x_n\} \to r$.

In a passage from some school slides I have written: consider an arbitrary $r \in Q$. Because $I$ is dense in $R$ there exists a sequence $\{x_n\} \subset I$ with $\{x_n\} \to r$. This seems ...
0
votes
2answers
42 views

Product metric spaces is again a metric space

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let: $$ d_2 ((x_1,y_1),(x_2,y_2)) = \left[d_X(x_1,x_2)^2 + d_Y (y_1,y_2)^2 \right]^{\frac{1}{2}} $$ for the points $(x_1,y_1)$ and $(x_2,y_2)$ in $X ...
4
votes
2answers
115 views

Bartle vs Rudin, which one is better for real analysis?

I'm in high school and I want to study real analysis, and I can choose between The elements of real analysis by Robert G. Bartle and Principles of mathematical analysis by Walter Rudin, so, from the ...
1
vote
2answers
133 views

on a recursive sequence (exercise 8.14 Apostol).

The exercise asks to prove convergence and find the limit of the sequence:$$a_n= \frac{b_{n+1}}{b_n},\text{ where } b_1=b_2 =1, b_{n+2} = b_{n} + b_{n+1}. $$ It also gives a hint: Show that $ \ ...
1
vote
0answers
40 views

What is the most “powerfull” method to prove a sequence is increasing or decreasing?

Given a sequence $a_n$ defined in a recursive manner, the methods I know to prove if the sequence is increasing are: 1) observe if $a_{n+1} - a_n > 0 \ \forall n.$ 2) take $\frac{a_{n+1}}{a_n}$ ...
1
vote
1answer
68 views

general topology (self learning)

Hi everyone I'd like to know if the following is correct. I'd appreciate any suggestion. Thanks in advance. From Dudley´s book: Let $A_n$ be the set of all the integers greater than $n$. Let ...
1
vote
2answers
53 views

on the exercise 8.10 Apostol. (limit of sequence)

The exercise states: prove that the limit of the sequence $$a_{n+2}=(a_na_{n+1})^{1/2} \ where \ a_1 \ge 0, a_2 \ge 0 $$ is $L = (a_1a_2^2)^{1/3}$ The solutin says: $$Let \ b_n = ...
4
votes
1answer
67 views

Exam exercise on sequence $a_n = \sin(n)$ [duplicate]

Prove that the sequence $a_n = \sin(n)$ cannot converge when $n \rightarrow \infty $ I tried to find two subsequences that converge to different values but I am having trouble with the fact that $n ...
0
votes
2answers
81 views

Proving $(1-x)^n \geq 1 - nx $

$(1-x)^n \geq 1 - nx\,\, $ If i expand the left side of the inequality with the binomianl coefficient formula I obtain: $1-nx + {n \choose 2}x^2 - {n \choose 3}x^3 ... $ now I see where the $1-nx$ ...
4
votes
1answer
57 views

On a recursive sequence (exercise 8.9 Apostol)

The exercise states: show convergence of the sequence ${a_n}$ knowing that: $$|a_n| \le 2, \ \ \ |a_{n+2}-a_{n+1}| \le \frac{1}{8}|a_{n+1}^2 - a_{n}^2|.$$ The solution states: $$|a_{n+2}-a_{n+1}| ...
2
votes
1answer
84 views

exam exercise on Series problem.

The exercise states: Does the series $$\sum\limits_{n=1}^\infty \int_{0}^1 \frac{x^n dx}{x+1}$$ converge? The solution states as the first step: $$I_n =\int_{0}^1 \frac{x^n dx}{x+1} $$ then $ ...
2
votes
1answer
38 views

Testing convergence of the series of $n^p((n-1)^{-1/2}-n^{-1/2})$

Exercise 8.15 (l) of Analysis by Apostol states: Test for convergence: $$\sum\limits_{n = 2}^\infty n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt{n}} \right)$$ The solution I have states as the first ...
2
votes
1answer
117 views

$[0,1]^{[0,1]}$ is separable

This is from Dudley´s book: Let $I:=[0,1]$ with the usual topology. Let $I^I$the set of all the functions from $I$ to $I$ with the product topology. a) $I^I$ is separable. Hint: Consider function ...
0
votes
1answer
29 views

Showing a set is closed, question from real analysis

Let $\mathbb{X}=\{1,2,3\}$ and let $\mathbb{P}$ be the set of all probabilities on $\mathbb{X}$. $\,\,$ Let, $V:\mathbb{P}\to \mathbb{R}$ be defined as $V(p)=(1+p_1)^2+(2+p_2)^2+(3+p_3)^2$. Show that ...
1
vote
1answer
96 views

What is the (rigorous) reason that the derivative of $|x|$ does not exist at $x=0$?

Let $g=|x|$. Then, the derivative at $c=0$ is given by: $$ g'(0) = \lim_{x \to 0} \frac{|x|}{x} $$ which is either $+1$ if $x$ comes from the positive $x$-axis or $-1$ if $x$ comes from the negative ...
1
vote
0answers
56 views

Suggested book for self study.

I have a degree in Financial Risk Management, and did 4 semesters of calculus and analysis(but that was about 10 years back), with most of my other efforts going toward Mathematical Statistics and ...
1
vote
0answers
25 views

net of indicator functions

Hi everyone I was reading Dudley's book and I'm having problems with the following. If $X$ is uncountable, show that there is a net of indicator functions of finite set converging pointwise to the ...
1
vote
1answer
57 views

Question about limit points in relation with continuity and functional limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have the feeling that the author is being careless about limit points in his theorems or I am not understanding ...
1
vote
2answers
62 views

Proof of Divergence Criterion for Functional Limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I don't understand corollary 4.2.5 on page 107. To be more specific, let me first write down the theorem that precedes ...
1
vote
1answer
32 views

Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
3
votes
2answers
132 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
0
votes
1answer
48 views

Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...
2
votes
2answers
40 views

exercise on pointwise convergence of an (easy) function.

Exercise 6.2.5. Taken from understanding analysis of Stephen Abbott For each n $\in N$, define $f_n on \ R$ by $$f_n(x) = \begin{cases} 1, & \mbox{if} \ |x| \ge 1/n \\ n|x|, & \mbox{if} \ ...
1
vote
2answers
127 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
0
votes
1answer
74 views

Monotonic Functions and Uniform Convergence

The following is a proof from "Heavy-Tail Phenomena" by Resnick (2007). I have some questions about the proof. (2.3) seems to be an identity. The left side the global sup over $[a, b]$ and hence ...
2
votes
1answer
46 views

Topology of a nested sequence of subsets

Hi everyone I'd like to know if the following proof is correct, I think so. And also if there is a more direct approach without the many subcases. Thanks in advance Let $X$ be an infinite set, and ...
2
votes
1answer
39 views

Uniform convergence of $xe^{-nx}$

Does the sequence $(f_n)$ on $[0, \infty)$ given by $ f_n(x) = > xe^{-nx} $ converge uniformly? This is from Bartle's Elements of Real Analysis. I've already proven that the sequence is ...