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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12 views

What is a cluster point of this sequence?

I'm trying to make sure I have correctly understood Bolzano-Weierstrass, which states that every compact subset of $\mathbb{R}^{n}$ is sequentially compact, which means that if $A$ is a compact subset ...
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1answer
10 views

Continuous map from $L^r(\Omega)$ to $L^s(\Omega)$.

The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory: Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that (i) $g \in C(\overline{\Omega} ...
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36 views

Help understanding theorem proof

So this is my first semester taking a Real Analysis class. We are using the book Introduction to Analysis by Gaughan 5th ed. This is my first real Math class and I'm really excited but I am having ...
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1answer
28 views

Function between two metric spaces?

I need to come up with: two metric spaces ( X , d ) and ( Y , p ) A continuous function f: X → Y A Cauchy sequence {xn} in X that isn't mapped to a Cauchy sequence in Y My idea was to make ...
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1answer
13 views

Proof a real functional is continuous in $C_{[a,b]}$ (verification)

I wish to have some feedback on the following proof of the claim below, either if it is correct, what to fix, or other suggestions. Claim: Let $\psi :[0,1] \times \mathbb{R} \to \mathbb{R}$ be a ...
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2answers
39 views

Prove that $\lim_{x \to +\infty} \frac{f(x)}{x} = L$ if $\lim_{x \to +\infty} [f(x+1) - f(x)] = L \space$

Let $f:[0, +\infty) \rightarrow \mathbb{R} $ be a bounded function in each bounded interval. If $$\lim_{x \to +\infty} [f(x+1) - f(x)] = L$$ then $$\lim_{x \to +\infty} \frac{f(x)}{x} = L$$ I tried ...
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13 views

Show that $Gr(f)$ is compact

Let $A \subset \mathbb{R}^n$ a compact and $f : A \to \mathbb{R}^m$ a continuous function. Let the graph of $f$ $$Gr(f) = \{(x,f(x) : x \in A)\}.$$ Show that $Gr(f)$ is compact. My proof : ...
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1answer
16 views

Definition of continuity up to the boundary

Let $\Omega \subset \mathbb{R}^n$ be open and bounded. What does it mean $f\in C(\bar{\Omega})$, i.e. what does it mean $f$ to be continuous at $x \in \partial \Omega$, maybe $$\forall \epsilon >0 ...
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27 views

How to relate the following function with Thomaes function

I am wondering how I can go about comparing the following functions $f(x)=\begin{cases} 1/n &\text{if $x=\frac{1}{n}$} \\ 0 &\text{else} \\ \end{cases}$ On the interval $[0,1]$, with the ...
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1answer
21 views

Uniform Convergence of Series tends to $f(x)$

The question is to prove that: $$\frac{f_1(x)+\cdots+f_n(x)}{n}$$ tends to $f(x)$ uniformly on $E$, as $n$ tends to infinity. I am not sure how to do this generically without an actual sequence. ...
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27 views

Does a bounded sequence in $C^k$ have a convergent subsequence in $C^{k-1}$

Let $K \subset \mathbb{R}^n$ be compact, let $k \in \mathbb{N}$. Let $\{f_n\} \subset C^k(K)$ be a bounded sequence w.r.t $C^k$ norm. Does it have a convergent subsequence in $C^{k-1}(K)$. If so how ...
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1answer
37 views

Splitting integral !!

i have this simple question that make me really confused : let $\phi$ a smooth function with compact support and $p>0,t\ge 0$ and : \begin{equation} U_p= \left\lbrace \begin{array}{ccc} 0 & ...
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2answers
53 views

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero?

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero? I was talking to a friend about it but neither of us were able to come up with anything. We're not sure if $\pi$ is essential ...
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2answers
61 views

Is there a nice open set proof that multiplication is continuous?

For students in a first course in analysis or topology, proving that certain function are continuous can be very tricky. However, some proofs which are difficult for students to prove using the ...
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1answer
39 views

Calculate the product limit by using elementary high school techniques

$$\lim_{n\to\infty} \prod_{i=2}^n(1-{1\over{i+1 \choose 2}}) $$ This is a problem i have encountered in one of my textbooks.Solve it by using high school methods for real analysis.
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19 views

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,…$

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,...$ I've tried to isolate n but its impossible to do.
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23 views

Calculus of integral over balls

I have this: $$\int_{B_{r\lambda_n/2}(y_n)\cap[B_{\lambda_n R}(0)\setminus B_{\lambda_n r}(0)]} x_1 |\nabla u_n|^p dx+\int_{B_{r\lambda_n/2}(y_n)\setminus[B_{\lambda_n R}(0)\setminus B_{\lambda_n ...
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1answer
57 views

Solve $x^2 = 2^x$. [duplicate]

One can see that the solutions are $x=2, 4$ and $x=-0.77$(approximately) seen from the graph. I am posting this to find if there is a way to solve this and find solutions like polynomial equations. ...
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1answer
32 views

If $f$ is differentible at a point $x \in [a,b]$, then $f$ is continuous at $x$.

Proof. As $t\rightarrow x$, we have, by Theorem 4.4, (Baby Rudin, p.104) $f\left( t\right) -f\left( x\right)$ = $\dfrac {f\left( t\right) -f\left( x\right) } {t-x}\cdot \left( t-x\right) \rightarrow ...
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2answers
27 views

Series of functions converge uniformly but sequence of functions does not

Given $a>1$ and $$f_{n}(x)=\frac{1}{1+n^{a}x^{4}}$$ I'm asked to show that for any $\delta >0$, the series of functions $\sum f_{n}(x) $ converges uniformly for $\{x \in \mathbb{R} | |x| \geq ...
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26 views

Cluster points and the sequence 1,1,2,1,2,3,1,2,3,4,1,…

I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a ...
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1answer
33 views

Prove that if $f:[a,+\infty [\longrightarrow \mathbb R$ is uniformly continuous, then $\lim_{x\to +\infty }f(x)=+\infty $ [on hold]

I have to show that if $f:[a,+\infty [\longrightarrow \mathbb R$ is uniformly continuous, then $\lim_{x\to +\infty }f(x)=+\infty $. I spend very much time on it, and I can't conclude. How can I find a ...
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3answers
98 views

Russell's paradox question

Tao's analysis book uses following example for Russell's paradox: $$P(x) \Longrightarrow `` x\text{ is a set, and }x \notin x"\\ \Omega := \{x : P(x)\text{ is true} \} = \{x : x\text{ is a set and }x ...
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3answers
18 views

Finding disc of convergence

Find the disc of convergence $$\sum_{n=0}^\infty z^{n^{3}}$$ I have applied the ratio test but I can not seem to come up with a conclusion.
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1answer
17 views

Bounded sequences without a convergent subsequence converging in a different metric

Given the metric $$d(x,y)=\text{min}\{1,|x-y|\}$$ on $\Bbb{R}$.There is a bounded sequence in $(\Bbb{R},d)$ without a convergent subsequence. Prove that a sequence in $(\Bbb{R},d)$ converges iff it ...
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40 views

Bound the first derivative of the following function: $f(x)=g(x)+h(x)$

Consider a decreasing function $f(x)$, resulting from the sum of two other decreasing functions: $f(x)=g(x)+h(x)$. All these 3 functions are positive. In addition, we have $g(0)=h(0)=1$. Further, we ...
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1answer
16 views

Find a power series centered at the origin that satisfies the Bessel

Find a power series centered at the origin that satisfies the Bessel differential equation $$zf''(z)+f'(z)+zf(z)=0$$ with initial condition $f(0)=1$. Show that this series converges for all z in C. I ...
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1answer
26 views

$A$ and $B$ be non-empty bounded set of real numbers, give a counter example to the following.

Assume $A \cap B \neq \emptyset$. Find a counter-example to the claim: $\sup(A \cap B) = \min\{\sup(A), \:\sup(B)\}$ I cant seem to find a counter example to the above claim, can anyone provide a ...
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2answers
46 views

How can I show that this function is discontinuous at the point $x=1$?

Suppose you had the function $$ f(x) = \; \text{ the integer part of } x $$ I wish to show that this is not continuous at the point $x=1$, which I will try to do by showing that $\lim_{x \rightarrow ...
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0answers
21 views

Find multiple roots of a three-dim system

Consider the three equations $$ y-x^2=0,\quad z+xy=0,\quad -y-z+x^2-xy+y^2+z^2-x^4=0. $$ How can I find multiple roots of this? Is it allowed to reduce the system as far as possible and then to find ...
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0answers
20 views

Parameterization which is closed under addition

Suppose $\beta_1(t)$ and $\beta_2(t)$ are two parametric curves defined on $[0,1]$. Let $\beta_1^*(t)$ and $\beta_2^*(t)$ are two re-parametrized of the above curves. Now, I looking for a ...
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1answer
51 views

How to prove that for all $k\in\mathbb N$, $h(kx)=kh(x)$ and $h(x+y)\le h(x)+h(y)$?

Suppose $X$ is a commutative monoid and $f:X\to\mathbb R\cup\{\infty\}$ a function and $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ ...
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21 views

What values of $b$ such that $f_n(x)=b\cos\left(\frac{x}{n}\right)$ converges uniformly?

For what values of $b$ does the sequence of functions: for each $n\in\mathbb{N}$, let $$f_n(x)=b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1]$$ converge uniformly in the space $C[0,1]$ equipped ...
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2answers
54 views

A sequence of functions converges in $C[0,1]$ iff it is Cauchy? Is it pointwise or uniform convergence?

In my notes, there is this theorem: A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy. I understand that this theorem applies to all complete metric spaces, not just to $R^n$. ...
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2answers
74 views

Theorem 2.43 in Baby Rudin: How to understand the proof?

Here's Theorem 2.43 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $P$ be a non-empty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Here's the ...
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1answer
49 views

Check the proof of $||x||^2$ is not a norm

Show if $f$ is a norm: For $\mathbb{R}^n$, Define $f: \mathbb{R}^n \rightarrow \mathbb{R} $ by $ f(x) = \|x\|^2$ =$\sum_{n} x_n^2 $ I tried to solve if $f$ satisfies the three properties of a norm: ...
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30 views

Why does this follow from the triangle inequality?

Proving that differentiability implies continuity.
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26 views

Which metric is used in this limit

Rudin's Principles of Mathematical Analysis(3rd ed) says on page 53 that 'If $\{p_n\}$ is a sequence in $X$ and if $E_N$ consists of the points $p_N,p_{N+1},p_{N+2},\dots$, it is clear from two ...
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24 views

Prove that a function Riemann integrable is [on hold]

I know there are questions like these, but I still don't understand how to prove it. Question 1 Do I use an epsilon proof or do I use the method of showing that $$\sup L(P, f) = \inf U(P,f)$$ ...
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13 views

Lebesgue measure of region under curve

Let $(X,\Sigma,\mu)$ be a $\sigma$-finte measure space and $f \in L^+(X,\Sigma)$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. Theorem: Define the area under the graph of $f$ to be ...
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52 views

About the gradient of a function in $H^{1}(\Omega)$

let $\Omega \in \mathbb{R}^{n}$ a bounded domain and $u \in H^{1}(\Omega)$ a real function. In the Leoni's Book - A First Course in Sobolev Spaces, the author define $\nabla u = (D_{1} u,\dots, ...
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1answer
18 views

How to prove that the steeple function is not uniformly convergent?

In class we encountered this function $$f_n(x)=\begin{cases} n^2x, & 0 \leq x \leq 1/n\\ 2n - n^2x, & 1/n \leq x \leq 2/n\\ 0, & 2/n \leq x \leq1 \end{cases}$$ The prof said ...
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23 views

About the definition of $L^{\infty}$ norm

Let $\Omega$ a limited domain in $\mathbb{R}^{n}$, the space $L^{\infty}(\Omega)=\{f: \Omega\to\mathbb{R} $ measurable $; ||f||_{L^{\infty}(\Omega)}<\infty\}$. Then if a function $f \in ...
3
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1answer
48 views

Prove if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges. No, $a_n, b_n$ are not necessarily ...
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3answers
18 views

Prove the existence of limit of certain sequences.

Problem: Let $0<a_1<b_1$ and $$a_{n+1}=\sqrt{a_n\cdot b_n},b_{n+1}=\frac{a_n+b_n}{2}.$$ Prove that $\{a_n\}$ and $\{b_n\}$ converge to some limit. Attempt: By induction and AM-GM, I can show ...
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1answer
54 views

To refute : a function with one discontinuity point is integrable on $\left[0, 1\right]$

If, let $f: \left[ 0, 1\right] \to \mathbb{R}$ continuous with only one discontinuity point is integrable on $\left[ 0, 1 \right]$. I think this is false but I can't find a example that contradicts ...
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0answers
17 views

Uniformly continuous of a function [duplicate]

Let $f:\mathbb{R}\to \mathbb{R} $ be an uniformly continuous functions, how to prove that there are $a, b $ such that $|f(x)|\leq a|x|+b$ thanks for any suggestions
1
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3answers
40 views

application of the inequality $\|fg\|_1 \leq \|f\|_p\|g\|_q$

application of the inequality $\|fg\|_1 \leq \|f\|_p\|g\|_q$ where $1/p + 1/q = 1$ I know this is a straight application of the inequality, but how am I assured that the integral of ...
0
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0answers
17 views

Verifying a startegy to prove convexity on partial domain

Assume you have the multivariate function $$f(x_1,x_2,..,x_n)$$ where: $x_i>0 \forall i$, and $\sum_i x_i = 1$. I need to show that $f$ is a convex function. My plan is to show that it is ...
2
votes
1answer
31 views

Show $\cos(x^2)/(1+ x^2)$ is uniformly continuous on $\Bbb R$.

now here's how I did proceed. By definition a function $f: E →\Bbb R$ is uniformly continuous iff for every $ε > 0$, there is a $δ > 0$ such that $|x-a| < δ$ and $x,a$ are elements of $E$ ...