Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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2
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2answers
117 views

Totally disconnected implies nowhere dense

Why is it true that a totally disconnected space implies it is nowhere dense in the reals? I know that totally disconnected implies the component is a singleton, but how do we construct a nowhere ...
1
vote
1answer
39 views

Continuity of a composition of continuous functions

Suppose $f: \mathbb{R} \to \mathbb{R} $ is continuous at $x = 1 $ and $g: \mathbb{R} \to \mathbb{R} $ continuous at $y = f(1) $. Then $g \circ f $ is continuous at $x = 1 $ Attempt: Let $\epsilon ...
0
votes
1answer
30 views

Find the asymtotes to the function: $f(x)={1+ln|x|\over x(1-ln|x|)}$

Find the asymtotes to the function: $f(x)={1+ln|x|\over x(1-ln|x|)}$. I have difficulties with this one when i try to find the limits, i get indefinites that look like (infinity over infinity)over ...
5
votes
1answer
131 views

Is $\sum \sin(n^2)/n$ convergent?

Is the series $$\sum_{n\ge 1} \frac{\sin(n^2)}{n}$$ convergent? My thoughts so far: 1)This is an alternating series so the integration test does not work here. 2)The Weyl inequality roughly says ...
1
vote
0answers
57 views

Example of a function such that iterated integrals are equal

Is there any example of a function $f(x,y):[0,1]$x$[0,1]\to \mathbb R$ so that $\int_{0}^1\int_{0}^1f(x,y)dydx$ and $\int_{0}^1\int_{0}^1f(x,y)dxdy$ exists and are equal but $\int\int f(x,y)dydx$ does ...
2
votes
0answers
75 views

A proposition of Urysohn's Lemma in real analysis

I have a problem proving the following theorem, which appears in Evans' PDE text book: Let $K$ be a compact set in space $R^n$, and $K\subseteq \cup_{i=1}^{k} U_{i},k\geq2$, where $U_i$ are open ...
0
votes
0answers
33 views

An assertion in the proof of Riesz Representation Theorem: the Radon measure is zero over $f^{-1}(t_i)$

The Riesz Representation Theorem is this version: $\quad$Let $L:C_c(R^n,R^m)\rightarrow R$ be a linear operator, where $C_c(R^n,R^m)$ denotes the space of continuous map from $R^n$ to $R^m$ with ...
0
votes
1answer
31 views

$exp(-\frac{\sigma^2}{4a^2}) \leq \frac{a}{\sigma}$

How can I prove that $exp(-\frac{\sigma^2}{4a^2}) \leq \frac{a}{\sigma}$ for $\sigma > a$? I tried to proof it with the function $f(x)=-exp(-\frac{\sigma^2}{4a^2}) + \frac{a}{\sigma}$. But this ...
0
votes
2answers
46 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...
3
votes
1answer
52 views

exchange max and limit

Let $\lim_{n \rightarrow \infty} a_n = a$ and $\lim_{n \rightarrow \infty} b_n = b$ exist, then is it true that $\lim_{n \rightarrow \infty} \max \{ a_n, b_n \} = \max \{ a, b \}$? I couldn't find ...
2
votes
1answer
52 views

Construct a continuous non-monotone function

Construct a continuous non-monotone function $H\in BV(\mathbb{R})$ satisfying $H'\geq 1$ a.e. or, prove this is impossible. My intuition tells me that this is impossible. But I don't know how to ...
0
votes
1answer
30 views

Is the unit $f(x)=1$ an element of the algebra generated by a strictly monotone function?

What I am tring to prove: the algebra generated by a strictly monotone function is dense in $C\lbrack 0,1\rbrack$. What I have determined thus far: It seperates points but does it vanish? Consider ...
1
vote
1answer
62 views

Asymtotes to the the function: $y= \sqrt{{x^3-1\over x}}$ at x=0 there should be a vertical asymtote, but i don't know how to formally execute this..

Asymtotes to the the function: $y= \sqrt{{x^3-1\over x}}$ at $x=0$ there should be a vertical asymtote , but i don't know how to formally execute this. When i try to find the limit, i get the square ...
0
votes
1answer
44 views

Some Chain Rule Application

Let $F:\mathbb{R}^n\to\mathbb{R}^n$. Define $h:[0,1]\to\mathbb{R}^n$ by $$ h(s):=D_xF(a+sb)c. $$ Question: How should one interpret the RHS in $$ h'_k(s)=D_x^2F_k(a+sb)(b,c)\,? $$ Thoughts: I think ...
0
votes
2answers
27 views

Questions on open set Cantor Space

I am trying to show that each point in the Cantor space, $\prod_{i \in\mathbb{N}}\{0,1\}_i$ is a limit point. But I am confused about what is the open set of the Cantor space. Thanks!
0
votes
2answers
99 views

If $f$ is Lipschitz, $X_n$ converges in distribution and $d(X_n,Y_n)$ converges stochastically to $0$, then $\limsup_{n\to\infty}E[|f(X_n)-f(Y_n)|]=0$

Let $(E,d)$ be a separable metric space $X,X_n,Y_n$ be random variables with values in $E$ such that $(X_n)_{n\in\mathbb{N}}$ converges in distribution to $X$ and ...
0
votes
1answer
39 views

Application of Inverse Function Theorem

This is a seemingly easy exercise. Yet I am not sure if I am missing any finer details here as this is listed as one of the challenging problems on Dr. Epstein's (Upenn) course site for real analysis. ...
1
vote
1answer
40 views

What is the relationship between DTFT and continuous fourier transform?

As title says, what is the relationship between DTFT and continuous fourier transform? Let's say there is continious signal $f(t)$. Continuous Fourier transform convert this into $F(\omega)$. Now let ...
1
vote
1answer
28 views

Support of a convolution with the help of Titchmarsh theorem

I have to use Titchmarsh theorem in order to prove that : if $f\in L^1[-1,1]$, and $supp(f*f*f*f-f*f)\subset [-1,1]$ then $supp(f)\subset[-1/4,1/4]$. Does anyone have an idea ? Thank you.
1
vote
2answers
48 views

Differentiation of Power Series

Let $$f(x)= \sum_{n=0}^{\infty} \frac{x^n}{n!} $$ for $x\in \mathbb R$. Show $f′ =f$. Note: Do not use the fact that $f(x) = e^x$. This is true but has not been established at this point in the ...
0
votes
1answer
126 views

Find the radius of convergence of the Power series $1+z+\frac{z^2}{2^2}+\frac{z^3}{3!}+\frac{z^4}{2^4}+\dots$

Find the radius of convergence of the Power series $$1+z+\frac{z^2}{2^2}+\frac{z^3}{3!}+\frac{z^4}{2^4}+\frac{z^5}{5!}\cdots $$ Put the series in the form ...
10
votes
2answers
145 views

How to evaluate $ \sum\limits_{n=1}^{\infty} \left( \frac{H_{n}}{(n+1)^2.2^n} \right)$

Evaluate $$ \sum_{n=1}^{\infty} \left( \dfrac{H_{n}}{(n+1)^2.2^n} \right)$$ Where $H_{n}$ is the $n^{th}$ Harmonic Number, i.e., $H_{n} = \displaystyle \sum _{k=1}^n \frac{1}{k}$ I ...
4
votes
2answers
66 views

How to use the inverse function theorem to find a local inverse?

I have a function $F(x,y)=(x^2 - y^2, xy)$ and I need to show that it has an inverse. How do I find the inverse of this function using the inverse function theorem? I did this a while ago and now I ...
0
votes
1answer
20 views

Questions about total disconnectedness

I have come across the proof that the Cartesian product of totally disconnected sets is also disconnected in the following post. Product of totally disconnected space is totally disconnected? ...
0
votes
1answer
51 views

Is the function integrable?

Is the function $$f(x) = \begin{cases} \frac{1}{10^n}, & \text{if $x$ }\in(2^{-(n+1)},2^{-n}) \\[2ex] 0, & \text{if $x=0$} \end{cases}, f:[0,1]\to \Bbb{R}$$ integrable? Find $\int_0^{1}f$. ...
1
vote
1answer
41 views

Show that every continuous functions on closed interval is the uniform limit of a sequence of polynomial

If $0\notin [a,b]$ show that every continuous function $f$ on [a,b] is the uniform limit of a sequence of polynomials $q_n$ where $q_n = x^n p_n{(x)}$ for some polynomials $p_n$. Attempt so far: I ...
2
votes
1answer
29 views

Uniform Convegence of differentiable polynomials to differentiable functon

If $f$ is continuously different on $[0,1]$ show that there is a sequence of polynomials $p_n$ converging to $f$ such that $p_{n}^\prime$ converges uniformly to $f^\prime$. Scratch Idea Since $f$ is ...
0
votes
0answers
38 views

A problem in Weierstrass M-test proof

I am trying to prove Weierstrass M-test. My whole proof is as followed. I found a problem is that when I use the triangle inequality thm, it is actually cannot be implemented infinitely. I don't know ...
2
votes
4answers
80 views

Calculating value of integral of convolution using Fourier transform

Calcuate the integral $$I=\int_{-\infty}^\infty\frac{\sin a\omega\sin b\omega}{\omega\cdot \omega}d\omega.$$ First I noticed that $$\mathcal{F}(\mathbb{1}_{[-h,h]})(\omega)=\frac{\sin h ...
3
votes
3answers
85 views

How can I prove $\sum_{n=1}^{\infty} \frac{x^n}{n!}$ converges uniformly on $[-a,a], a>0$ but not on $\mathbb{R}$?

I'm looking to show that: $\sum_{n=1}^{\infty} \frac{x^n}{n!}$ converges uniformly on $[-a,a]$ for $a>0$, but does not converge uniformly on $\mathbb{R} $. How can I do this? My book ...
1
vote
1answer
44 views

How is this map injective?

Let $X$ be a (real or complex) vector space, let $X^{*}$ denote the vector space of all linear functionals defined on $X$, and let $X^{**}$ denote the vector space of all linear functionals defined on ...
8
votes
4answers
201 views

Evaluating $ \sum_{n=1}^\infty \frac{1}{n^2 2^n} $

Evaluate $$ \sum_{n=1}^\infty \dfrac{1}{n^2 2^n}. $$ I have tried using the Maclaurin series of $2^{-n}$ but it further complicated the question. Moreover, I have also tried taking help ...
2
votes
2answers
47 views

Prove $(x_n)=\sum_{k=0}^{n}\frac{x^k}{k!}$ is a Cauchy sequence

If $(x_n)=\sum_{k=0}^{n}\frac{x^k}{k!}$, prove that $(x_n)$ is a Cauchy sequence. Please give me a hint at how to proceed
3
votes
2answers
38 views

If $s_n=\sum_{k=0}^{n}a_k$ and $n\log n \ a_n\rightarrow 0 \ (n\to \infty)$ show $\frac{s_n}{\log n}\to 0 \ (n\to \infty).$

Let $s_n=\sum_{k=0}^{n}a_k$ and $n\log n \ a_n\rightarrow 0 \ (n\to \infty).$ Is it true that $\frac{s_n}{\log n}\to 0 \ (n\to \infty)?$
0
votes
1answer
56 views

Derivatives of Higher Order

Let $U \subset \mathbb{R}^n$ be an open set and $f : U \to \mathbb{R}$. The function $f$ is said to be in $C^k(U)$, if $f$ has all continuous partial derivatives up to and including order $k$ on ...
0
votes
1answer
46 views

Question About Continuity of an Antiderivative

Let $f(x)$ be a periodic function of period $T$ that is integrable over every finite interval. It is clear that $F(x)=\int_x^{x+T} f(x)dx$ is a constant. How can it be shown that $G(x)=\int_0^x ...
1
vote
0answers
68 views

Show that $\pi \cot \pi z = \frac{1}{z} + \sum\limits_{n=1}^{\infty} \frac{2 z}{z^2 - n^2}$

This question has been asked before, but we are supposed to prove for any $z \in \mathbb{C} \setminus \mathbb{Z}$ that $$\pi \cot \pi z = \frac{1}{z} + \sum\limits_{n=1}^{\infty} \frac{2 z}{z^2 - ...
0
votes
1answer
71 views

Question About the Proof that the Composition of a Riemann Integrable Function and a Continuous Function is Riemann Integrable

I have a question about the proof of the following theorem from Little Rudin: Theorem Suppose $f \in \mathscr{R}(\alpha)$ on $[a,b]$, $m<f<M$, $\phi$ is continuous on $[m,M]$, and ...
1
vote
2answers
40 views

Showing Function is Continuous

Let $f: \mathbb{R} \backslash \{2\} \to \mathbb{R}$ be the function given by $f(x) = \frac{2x^2+x-10}{3x-6}$. Let $g: \mathbb{R} \to \mathbb{R}$ given by: $$g(x)=\begin{cases}{f(x)} & \text{if } ...
1
vote
2answers
99 views

Prove that $[1,2]+[-2,-1]=[-1,1]$

$A+B=\{x\in\mathbb{R}: x=a+b$ for some $a\in$ A and $b\in B\}$ A=[1,2], B=[-2,-1], prove A+B=[-1,1] This is my attempt but my professor said it's not good enough. Proof: since all elements $x\in ...
3
votes
1answer
70 views

$L^2$ and $L^1$ space problem

For a $\sigma$-finite measure space $(\Omega,\mathscr{F},\mu)$, is $L^2\subset L^1$ always true?
8
votes
1answer
199 views

Show that $\lim_{n\to\infty}(6n)^{\frac16}a_n=1$ with $(a_n)$ such that $\lim_{n\to\infty}a_n\sum_{j=1}^na_j^5=1$

Show that $$ \lim_{n\to\infty}(6n)^{\frac16}a_n=1, $$ where $(a_n)$ is a sequence of nonnegative real numbers such that $\lim_{n\to\infty}a_n\sum_{j=1}^na_j^5=1.$ I recently got stuck on this ...
3
votes
2answers
62 views

If $S \subset T$ prove $\overline{S} \subset \overline{T}$ and $\text{int}(S) \subset \text{int}(T)$

Closure: $\bar{S} = \cap K$ where $K$ ranges over all closeds containing $S$. Interior: $int(S) = \cup U$ where $U$ ranges over all opens contained in $S$. My attempt for first part: Let $x$ be an ...
1
vote
2answers
70 views

Two fundamental questions about convexity of a function (number1)

The first question is as follows (see the second one): If $f$ is a convex function it is know that there is at most a single minimum. However the argument of the minimum is not guaranteed to be ...
2
votes
0answers
205 views

questions about Folland real analysis chapter 1 exercise

Here, E is a Lesbegue-measurable set on the real line. This is the exercise 30, 31 of p. 40 of Folland real analysis. I solved these problems when E is of finite measure, but the problem requires ...
0
votes
0answers
40 views

g(x) = 1/(1+x^2) continuous everywhere (is this solution correct) [duplicate]

how would you prove that g(x) = 1/(1+x^2) is continuous everywhere. I have the following: g is continuous at point a if for all ε > 0 there exists 𝛿 > 0 such that for all a in R, |x-a| < 𝛿 then ...
1
vote
0answers
65 views

proving point wise convergence but no uniform convergence on f

Let f$_n$: E → $R$ be continuous functions for 1 ≤ n ≤ N. Let a$_k$$^n$ be N convergent sequences of numbers and assume $\lim_{k \to inf}$ a$_k$$^n$ = a$_n$. Let f = $\sum_{n=1}^N$a$_n$f$_n$. I am ...
1
vote
0answers
48 views

Minor issue about the proof of the Cauchy Convergence Criterion on Understanding Analysis (Abbott)

I don't understand one small part of the proof of the following statement: If a sequence is a Cauchy sequence, then it converges. PROOF: Let $(x_n)$ be a a Cauchy sequence. Then it is bounded ...
1
vote
0answers
22 views

Showing that a function series is positive

I am stuck with the following problem. Let $w$ be the function defined on $\mathbb R$ by $$ w(x) = \left( 1-x^2\right) \exp(-x^2/2) \quad ,\quad x\in\mathbb R\;, $$ and a real number $\phi >1$ . ...
1
vote
2answers
36 views

Interior of a set $A$ is dense in $X$ iff $A$ is dense in $X$

I would like to check if the following statement is true: Interior of a set $A$ is dense in $X$ iff $A$ is dense in $X$ It seems to me that they are equivalent, but I couldn't prove or find a ...