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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1answer
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Oscillation of a function at a point

Why do we need to take open neighbourhoods around the point in consideration while defining oscillation of a function at that point? (We're working in R) For ref. Bartle & sherbert(introduction to ...
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1answer
23 views

Is $W^{1,2}_0$ a Hilbert space?

I came across the term $W^{1,2}_0$. Just a quick question on what the 0 means and is it a Hilbert space? I know $W^{1,2}$ is a Hilbert space. Thanks!
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2answers
30 views

prove limit of exponential function without concept of logarithm

The question is, prove that if a real number $x>1$, then $\lim_{n\to\infty}x^n = \infty$, where $n \in \mathbb N$, without using the logarithmic concept. I came up with a proof, but I'm not so sure ...
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1answer
29 views

Distinct real roots .

Problem : If $|\log(x)| - px = 0$ has three distinct real roots then the range of $p$ will be ? My attempt : I tried to see the problem graphically and made the graph. So I am able to see that ...
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12 views

Converting all arcs to polygonal arcs in a plane graph

I am trying to understand a proof on the conversion of arcs to polygonal arcs in plane graphs, in the book "Graphs on Surfaces" by Mohar and Thomassen. In the book, an arc joining two points $x,y \in ...
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1answer
23 views

Are these two definition of boundedness equivalent?

Definition 1: A set $S \subset M$ is bounded if $\forall x \in M, \exists r > 0,$ such that $S \subset B_r(x) = \{y \in M | d(x,y) < r\}$ Definition 2: A set $S \subset M$ is bounded if ...
0
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3answers
83 views

Show that $a_n = 1 + \frac{1}{2} + \frac{1}{3} +\dotsb+ \frac{1}{n}$ is not a Cauchy sequence

Let $$ a_n = 1 + \frac{1}{2} + \frac{1}{3} + \dotsb + \frac{1}{n} \quad (n \in \mathbb{N}). $$ Show that $a_n$ is not a Cauchy sequence even though $$ \lim_{n \to \infty} a_{n+1} - a_n = 0 ...
1
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1answer
36 views

Is this proof correct? Show $\mathbb{Q}$ is dense in $\mathbb{R}$

I like proof by contradictions in showing that $\mathbb{Q}$ is dense in $\mathbb{R}$. But I can't understand this one> https://math.dartmouth.edu/archive/m54x12/public_html/m54densitynote.pdf ...
2
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1answer
15 views

Show that $f$ is differentiable at point $x \not= (0,0)$ - $h(x) = (\sin ||x||)^p \cos \frac{1}{||x||}$

Let $p \geq 1$ and $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} (\sin \|x\|)^p \cos \frac{1}{\|x\|}, & \quad \text{if } \|x\| \not= 0 \\ 0, & \quad ...
0
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1answer
24 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
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1answer
22 views

Proof verification: Compact set has sup and inf

I was reading this post compact set always contains its supremum and infimum There was an answer reposted as follows: As $K$ is compact, we have that $K$ is bounded. So $\sup K$ and $\inf K$ ...
0
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1answer
21 views

Show that a sequence $\{s_n\}$ converges to $L$ if and only if the sequence $\{-s_n\}$ converges to $-L$.

I understand that both parts of this biconditional must be proven. If I assume that a sequence $\{s_n\}$ converges to $L$,then for every for every $ϵ>0$, there is some integer N where ...
2
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1answer
47 views

To show the $\epsilon-\delta$ definition for limits holds.

Question: Check if the following limit exists, if so show that the $\epsilon$ $\delta$ definition for limits holds. $$\lim_{(x,y) \to (1,2)} \frac{(x-1)^2(y-2)^2}{x^2+y^2-2xy-4y+5}$$ My answer: So ...
1
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0answers
34 views

Show that the sequence of functions $f_n(x)=xe^{-nx}$ for $x\in(0,1)$ converges pointwise to $0$

Is it enough to calculate $\lim_{n \rightarrow \infty} xe^{-nx}$ or should we analyze it more carefully?
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1answer
11 views

Is relative entropy with respect to a pmf a continuous function?

Is the relative entropy $D(p || q)$ with a fixed pmf $q$, continuous over $p$, where $p \in \{x \in \mathbb{R}^n: \sum_{i=1}^n x_i = 1 , x_i \geq 0 \}$?
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0answers
36 views

How to prove $\int_{-\infty}^{+\infty}|f(x+t) - f(x)|dx \to 0$ as $t\to 0$

Let $f\in L^1(-\infty, +\infty)$ and $g(t) = \int_{-\infty}^{+\infty}|f(x+t) - f(x)|dx$. How to show that $\lim_{t\to0}g(t) = 0$? Originally I thought that we need to use Dominated Convergence ...
2
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5answers
48 views

Prove that $f$ has a minimum

Let $f$ be a positive and continuous function in $[0,\infty)$, such that $\lim\limits_{x\to \infty} f(x)=2$. Prove that if $f(0)<2$, $f$ has a minimum in $[0,\infty)$. I am stuck in the ...
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0answers
22 views

Prove that $\sum_{n=1}^{\infty} \frac{[nx]}{n^2} $ is discontinuous at $x \in \mathbb Q$

$[x] := x - \lfloor x \rfloor$. I can prove that it is continuous at all irrational points using uniform convergence, but I don't know how to prove discontinuity in this case. I looked at this similar ...
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4answers
64 views

If $B\subset A$ and $f:A\to B$ is injective prove it's a bijection between $A$ and $B$

I want to show that if $B\subset A$ and $f:A\to B$ is an injective function then there's a bijection between $A$ and $B$. I believe my "proof" is wrong, I probably use too much "intuition" when I ...
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44 views

Real Analysis-Find the limit of this series [on hold]

Problem # 3 enter image description here Find the limit of this series.
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2answers
43 views

Question about proof: Uniform cauchy $\Rightarrow$ Uniform convergence

I have one quick question regarding the proof of a theorem contained in here : https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch5.pdf Theorem 5.13. A sequence $(f_n)$ of functions $f_n ...
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1answer
18 views

Growth rate of large sets

Suppose that ${a_k}$ is a real valued increasing sequence such that $$ \sum_{k=1}^{\infty} \frac{1}{a_k} = +\infty ,$$ i.e. $\{a_1,a_2,\ldots\}$ is a large set. If $\lim a_{k+1} - a_{k} = \infty$, ...
2
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2answers
50 views

Proof of $\lim_{h\to 0}\frac{f(a+ h)-f(a)}{h}=\ell$ when $\lim_{x\to a}f'(x)=\ell$

Suppose $f$ derivable on $\mathbb R$ and that $\lim_{x\to a}f'(x)=\ell$. Show that $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\ell.$$ The proof of my course goes like this: By mean value theorem, there is ...
2
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2answers
31 views

Several questions about Riesz–Markov–Kakutani representation theorem

This is a list of questions about Riesz–Markov–Kakutani representation theorem . 1)If $f\in L^1(\mu)$, is it true that $\phi(f)=\int_Xfd\mu$, where $\mu$ is given by the theorem? I am quite sure it ...
3
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5answers
39 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
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0answers
24 views

Can I use the Squeeze theorem with sequences of functions?

For example if I know that $f_n(x)\leq g_n(x) \leq h_n(x)$ for all $x$, then can I say that $ \lim_{n \rightarrow \infty} f_n(x)\leq \lim_{n \rightarrow \infty} g_n(x) \leq \lim_{n \rightarrow \infty} ...
2
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1answer
33 views

Maximal interval of solutions existence: $x'(t)=-x(t)+\sin x(t)+t^3$

$x'(t)=-x(t)+ \sin x(t)+t^3$ in $\mathbb{R}$ I consider the function: $$ f(t,x)=-x+\sin x + t^3 $$ $$\frac{\partial f}{\partial x}=\cos x-1$$ I see that: $$\left| \frac{\partial f}{\partial x} ...
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0answers
10 views

How can I find a mapping $G(x,y)$ [on hold]

How can I find a mapping $G:X\times X \rightarrow X $ (where $X\subset R$) such that $G(x,y)$ is non-increasing in arguments $x$ and $y$ and $G(x,x)=0$ for all $x\in X.$ Thanks
5
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2answers
36 views

Deducing the series expansion of $\arctan(x^2)$ via the series expansion of $\arctan(x)$ at $x=0$

Comparing the series expansion of $\arctan(x^2)$ and $\arctan(x)$ at $x=0$ it looks like one can take the result from $\arctan(x)$ and replace each $x$ with $x^2$ to deduce the series expansion of ...
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0answers
12 views

About Convergence of a series [on hold]

Is this series convergent? $$\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}n^2}$$ where $c_{n}^{N}$ is coefficient $x^{n}$ in chebyshev polynomial $T_{N}(x)$, i.e. ...
0
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0answers
20 views

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A $ such that $||x^{(N)}||\to \infty $ ...
0
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1answer
15 views

$X$ sequentally Compact implies that $X$ is complete

I was reading through Roydens book and there is one part that I don't understand. Here is the proof. Suppose $X$ is sequentially compact metric space, then $$X \text{ is sequentally compact}:= ...
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0answers
17 views

Question About Representation of Brownian Motion

In Stochastic Calculus with Financial Applications by J. Michael Steele he makes a specific representation of a Brownian Motion using wavelets. At one point he calculates the covariance, and he uses ...
1
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3answers
51 views

For $\int f < \infty$, the measure of the set of points where $f=\infty$ is zero.

I fear this question was already discussed here, but I was not able to find it. Please remove if it is a duplicate. Prove: For a function $f\geq 0$, if $\int f < \infty$, then the measure of ...
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1answer
12 views

Finding First Integrals in the case $2xy u_x - (x^2+y^2) u_y =0$

Good day, As described in the title, I want to find two First Integrals (FI) to the PDE $$2xy u_x - (x^2+y^2) u_y =0$$ Of course, $u$ is a FI and the solution of the PDE ist $u(x,y)=u_0$. But I want ...
0
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2answers
37 views

Maximize the function $x+y$ on the closure of the unit ball

Find the maximum of $\{x +y : (x,y) \in closure [B(0,1)] \}$. Here $B(0,1)=\{(x,y) \in \mathbb{R}^2 : x^2+y^2 \leq 1\}$ I can't proceed in this as to how to maximize this value. Will it lie on ...
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1answer
36 views

Determine supremum and infimum of $\{x \in \Bbb R\,:\, x < 3/x\}$

Set $S := \{x \in \Bbb R\,:\, x < 3/x\}$. (a) Determine whether $\sup S$ exists, and determine its value if it exists. Justify your answer. (b) Determine whether $\inf S$ exists, and determine ...
2
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3answers
66 views

Calculate $\int_0^1 \ \int_0^1 \ x \sin \lvert x^2-y^2 \lvert \; dx \; dy$

$$\int_0^1 \ \int_0^1 \ x \ \sin \lvert x^2-y^2 \lvert dx \ dy $$ $$\int_0^1 \frac{1}{2} \Big[ \sin \lvert x^2-y^2 \lvert \Big]_0^1 \ dy= \int_0^1 \frac{1}{2} \Big( \sin \lvert 1-y^2 \lvert - ...
4
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5answers
56 views

About inner products, norms and metrics

Do these three kinds of vector spaces, those with an inner-product, those with a norm and those with a metric, are the same sets of vector spaces? At least for finite dimensional vector spaces all of ...
2
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1answer
26 views

$\sigma$-algebra produced by a subclass of a class.

im studying the book 'probability & measure' by Patrick Billingsley. in chapter 2 there's an exercise 2.9 say's: show that: If $B\in\sigma(A)$, then there exists a countable subclass $A_B$ of $A$ ...
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1answer
33 views

How to establish convergence and find limit of the sequence$(n+1)^{1/\ln(n+1)}$ [on hold]

How to establish convergence and find limit of the sequence $(n+1)^{1/\ln(n+1)}$. I know its a stupid question but its kinda urgent so please help me out! Edit 1: It is urgent because I have to ...
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3answers
65 views

Show that f is measurable.

Let $a > 0, b \geq 0$ and the function $f: \mathbb{R} \to \mathbb{R}$ $$f(x) = \left\{\begin{matrix} 1, & |x| \leq a \\ b & |x| > a \end{matrix}\right.$$ show that it is measurable. ...
4
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2answers
59 views

Is $f(x)$ uniformly continuous?

$f:\mathbb R\to[0,\infty )$ is continuous such that $g(x)={(f(x))}^2$ is uniformly continuous. Then prove that $f(x)$ is uniformly continuous. I've tried to use the following method: $$ ...
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0answers
41 views

Prove that $f'(c)= \frac{2}{2+3(f(c))^2}$ for some $c$

Problem: $f: [0, 1] \to \mathbb{R}$ is continuous on $[0, 1]$ and differentiable on $(0, 1)$. ALso, $f(0)=1$ and $(f(1))^3+2f(1)-5=0$. Prove that there exists a $c \in (0, 1)$ such that $f'(c)= ...
3
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2answers
49 views

Construct $f: X\to Y$ such that $f(p)=p$

Let , $X=[-1,1]\times [-1,1]$ and $Y=\{0\}\times \left[-\frac{1}{2},\frac{1}{2}\right]$. Construct an example of a continuous map $f:X\to Y$ such that $f(p)=p$ for each $p\in Y$. I construct a ...
0
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3answers
25 views

How do I decide which convergence criterium to use and what to do if they don't work?

Take $$ \sum_{n=1}^\infty (-1)^nn^{1/n} $$ I just blindly tried (Ratio test) $$ \limsup_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| $$ and (Root test) $$ \limsup_{k\to\infty} \sqrt[k]{|a_k|} $$ ...
1
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1answer
13 views

Check if this is the example of $x$ as a limit point of $C$ but ($x_t$) does not converge to $x$.

Let ($x_t$) be a sequence in a metric space, and let $C$ be the range of ($x_t$). I want to give an example in which $x$ is a limit point of $C$ but ($x_t$) does not converge to $x$. Here's my ...
2
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0answers
18 views

Boundary of a $k-$Cell Definition

In the book I am reading, the Boundary of a $(k+1)-$cell $\varphi$ is defined to be the $k-$chain $$\partial\varphi=\sum_{j=1}^{k+1}(-1)^{j+1}(\varphi\circ \iota^{j,1}-\varphi\circ \iota^{j,0}) $$ ...
1
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1answer
40 views

prove that if $a>1$ , then the limit of $\frac{a^n}{n}$ is $\infty$ as n goes to $\infty$

prove that if $a>1$ , then the limit of $\frac{a^n}{n}$ is $\infty$ as n goes to $\infty$ I was trying to use the binomial theorem to replace $a$ with $1+k$, but then that $n$ on the denominator ...
1
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1answer
35 views

prove the following limit without evaluating the integrals explicitly

$\lim_{n\to \infty}\int_1^2 \ln^n(x) \, dx=0$ $\lim_{n\to \infty}\int_2^3 \ln^n(x) \, dx=\infty$ I know for #2 I need a lower estimate for the integral that is large to show that the limit is ...