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
36 views

Prove that $||a|-|b||\leq |a-b|$ for all real numbers

Prove that $||a|-|b||\leq |a-b|$ for all real numbers I was thinking divide it into $a\geq b$ and $a<b$, but then I realized I need to include situations when they are greater than zero and less ...
1
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1answer
23 views

Show that $S^1$ is complete

Define: $S^1 = \{(x,y): x^2+y^2 = 1\}$ Then wish to show $S^1$ is complete Attempt: Let $(z_n)$ be a Cauchy sequence on $S^1$, $(z_n) = (x_n, y_n)$ Then $S^1$ is complete if $\forall ...
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1answer
19 views

if $-a\leq b\leq a$, then $|b|\leq a$

if $-a\leq b\leq a$, then $|b|\leq a$ I started by showing that $-|b|\leq b\leq |b|$, but then I can only see that $a$ can be equal to $|b|$, how do I show $|b|\leq a$?
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2answers
34 views

Prove that if $\mu (A) = \nu(A)$ for all $A \in s$, then this also holds for all $A \in M(s)$

Let $s$ be a collection of subsets of $X$. Assume that $\mu$ and $\nu$ are two measures on $M(s)$. Prove that if $\mu(A) = \nu(A)$ for all $A \in s$, then this also holds for all $A \in M(s)$, i.e., ...
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2answers
48 views

Is every set in $S$ measurable with respect to the outer measure induced by $\mu$

Let $S$ be a collection of subsets of $X$ and $\mu : S \to [0, \infty]$ a set function. Is every set in $S$ measurable with respect to the outer measure induced by $\mu$ Here is how we defined outer ...
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1answer
21 views

How can I show the points of continuity of the following function

How can I show the points of continuity of the following function $$f(x) = \begin{cases} 2x, & \text{if $x \in \Bbb Q$} \\[2ex] x+3, & \text{if $x \in \Bbb I$ } \end{cases}$$ I am having ...
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1answer
53 views

Homeomorphism between $\mathbb{R}$ and $\mathbb{Q}$ - why does cardinality matter?

When I look up why $\mathbb{R}$ and $\mathbb{Q}$ are not homeomorphic, almost all the answers just say something along the line of "Because, Cardinality" and then ends there. Can someone provides ...
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0answers
24 views

Inner measure of a set

My question is problem 15 of chapter 3 of Wheeden and Zygmund which states: If $E$ is measurable and $A$ is any subset of $E$, show that $m(E)=m_{*}(A) + m^{*}(E-A),$ where $m_{*}$ and $m^{*}$ denote ...
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0answers
31 views

Fourier coefficients of the Gaussian.

I would need to find the fourier coefficient of this gaussian for a problem. I'm now stuck with this integral, \begin{equation} c_{n}=\int_{-1}^{1}e^{\frac{x^{2}}{2}}\left(\cos\left(\pi ...
4
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2answers
45 views

Prove that $\lim\limits_{x\to\infty} f'(x)=0$

Let $f$ be a function in $(0,\infty)$ such that $f'(x)$ exists. In addition, $\lim\limits_{x\to \infty} f'(x)=L$ (finite) and $f(n)=0$ for every $n \in \Bbb N$. Prove that ...
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1answer
27 views

Rudin's RCA, Chapter 2 Definitions

I am currently reading Rudin's RCA, and I have some questions about a particular definition he uses in chapter 2: The following passage is taken from Rudin's RCA, page 47, section 2.15: "A measure ...
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1answer
37 views

Why does $x_n^2$ converge to $x^2$ if $x_n$ converges to $x$?

Let $x_n$ be a convergent sequence converging to $x$ Then claim $x_n^2$ converges to $x^2$ I wish to use the definition to show this is the case. Recall $x_n \to x$ iff $\forall \epsilon > 0, ...
2
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2answers
57 views

Show $\forall \delta > 0, \exists n \in \mathbb{N}$ such that $\frac{1}{n} < \delta$

The question is in the title, but I have no idea how to solve it, so a few hints would be appreciated, thanks.
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0answers
21 views

Show that $J_n(x)$ satisfies Bessel equation $ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $

Here is the definition of the Bessel function I am starting with a definition as an integral. $$ J_n(x) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i n t - x \sin t} \, dt $$ Essentially we have computed ...
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5answers
38 views

Prove $a_t \rightarrow x$ using the Betweenness Property

Prove that for any $x \in \Bbb R$ there is a strictly increasing sequence ($a_t$) in $\Bbb Q$ such that ($a_t$) converges to $x$, (i.e. $a_t \rightarrow x$) I want to prove this using the Betweenness ...
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2answers
40 views

Proving Lipschitz continuity of a piecewise function

Define a locally lipschitz and nonnegative function $f\colon\mathbb{R}^n\to\mathbb{R}$. Let $M\in\mathbb{R}^{n\times n}$ and $\eta>0\in\mathbb{R}$. Consider the function ...
1
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0answers
28 views

E is measurable, then measure of E is the sum of the inner measure of a subset of E and the outer measure of the complement of the subset in E

If E is a measurable and A is any subset of E, show that $|E|=|A|_i+|E-A|_e$ where |E| is the measure of of E, $|A|_i$ is the inner measure of A, and $|E-A|_e$ is the outer measure of $E-A$. I have ...
3
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1answer
41 views

Prove $\lvert q\alpha - p \rvert \gt \frac cq $ using real analysis

Assume that $\sqrt 2$ exists, and let $a=\sqrt 2$. Prove that there exists a number $c > 0$ such that for all integers $q,p$, and $q\neq0$ we have $$\lvert q\alpha - p \rvert \gt \frac cq $$ Note: ...
4
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4answers
93 views

Find minimal integer $n>1$ for which $2^n > n^{1000}$.

Find the minimum integer $n>1$ for which $2^n > n^{1000}$. I have taken the $log$ on both sides, but not reached any result. I would appreciate if anybody will solve it accurately. Thanks in ...
5
votes
7answers
202 views

Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis

I studying in Real Analysis 2, but I have no idea how to solve this problem. My guess is to use Mean Value Theorem or a similar theorem? Could any one help me? Thanks.
1
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1answer
22 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
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3answers
33 views

Prove that the solution for $y'=y^3(1-\tan^2(\arcsin(y)))$ , $y(0)= {\pi \over 8}$ , is bounded.

I got this problem to prove, and I assume I need to use the existence and uniqueness theorem for non-linear ODE's, so I set $y' = f(x,y)$ and differentiating in respect to $y$ gives: $f_y(x,y)$. And ...
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1answer
22 views

Differentiable any finite number of times

Does there exist a pathological function which is differentiable any finite number of times as one wishes, but is not differentiable an infinite amount of times? Is it reasonable for such function to ...
0
votes
1answer
31 views

A faithful positive Radon measure

Let $X$ be a locally comapct and Hausdorff space. We say a positive Radon Measure on $X$ is faithful if $$0\leq f ~~~,~~~\int fd\mu=0\rightarrow f(x)=0 ~~\forall x\in X$$ Q: True or false: If there ...
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0answers
22 views

Support of a Radon measure

Let $X$ be a locally compact and Hausdorff space. For a given Radon measure $\mu$ on $X$, the support of $\mu$ is the smallest closed subset of $X$ with $|\mu|(X)=\lVert\mu\rVert$ (where $|\mu|$ is ...
1
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1answer
15 views

Nonlinear operator sends bounded set to relatively compact set

Consider $g$ a continuous function on $[a,b]\times\mathbb{R}$, and let $z_0\in\mathbb{R}$. Define the (nonlinear) operator on $C[a,b]$: $$Mv(x)=z_0+\int_a^x g(t,v(t))\,dt$$ for $x\in[a,b]$. Prove ...
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2answers
32 views

Lp space Inclusion Examples

I proved for a bounded set $\Omega$ and $1 \leq p \leq q \leq \infty$ that $L^{q}(\Omega) \subset L^{p}(\Omega)$. What is an example that would show strict inclusion, $ p<q$, and false if $\Omega$ ...
2
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0answers
27 views

Poincare Inequality in n-dimensions

I am trying to prove the Poincare Inequality on a n-dimensional box. That is a domain $ \Omega = (0,1)^n$ for $f(x) \epsilon H^{1}_{0}(\Omega) $, show there exists a constant $C$ such that ...
1
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2answers
44 views

Functions between metric spaces (and how they relate to closures of sets)

Let $(X,d)$ and $(Y , p)$ be metric spaces. Prove that if $f : X \to Y$ is continuous, then for any set $A\subset X$ with closure $\overline{A}$ we have $f(\overline{A})\subset \overline{ f(A) }$ ...
2
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1answer
26 views

A subspace of a mapping space?

We have a set $$ M=\{f:\mathbb{R} \rightarrow \mathbb{R}\mid f(1)>0\}\;.$$ I have never encountered this kind of set before. I assume it is correct to say that $M$ is a subspace of a mapping ...
0
votes
1answer
49 views

Why is continuity needed to substitute value of derivative inside Riemann-Stieltjes Integral?

Given $f$ increasing on $[a,b]$, $g(x)\in R(\alpha)$ on $[a,b]$, $\alpha \in C([a,b])$ and $\alpha \in BV([a,b])$ $$ \beta(x)=\int_a^xg(z)d\alpha(z) \text{ on [a,b]} $$ Why is the additional ...
0
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1answer
43 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
3
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1answer
38 views

Determining an upper bound

I have a function $$f(\lambda)=n\ln(1-p+pe^{\frac{\lambda}{n}})-\lambda p$$ I need to prove that $$f(\lambda)\leq \frac{\lambda^2}{8n}$$ using Taylor expansion. I have used the taylor expansion for ...
0
votes
1answer
37 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
1
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1answer
43 views

Limit using definition

I am trying to find $$\lim\limits_{x \to \infty} n-\sqrt{n^2-4n} $$ using the definition of a limit. I have tried to multiply top and bottom by $n+\sqrt{n^2-4n} $ giving $\frac{4n}{n+\sqrt{n^2-4n}} ...
0
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2answers
28 views

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n\}$ is closed

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n \}$ is closed. I had to show it is compact, and I am done showing it is relatively compact, but now I am stuck showing it is closed. ...
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2answers
55 views

Show that $f$ is uniformly continuous - $\lim_{\|x\| \to \infty}f(x) = c$

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a continuous function. We suppose that there exist $c \in \mathbb{R}^m$ such that $$\lim_{\|x\| \to \infty} f(x) = c.$$ Show that $f$ is uniformly ...
0
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2answers
27 views

Compact set of a set $K$

Show that $K := \{x \in C[0,1] : x(0) \in [-3,4], |x(t)-x(s)| \leq d |t^2-s^2|, \forall t,s \in C[0,1]\}$ is compact. I already know that $C[0,1]$ is a compact set. So it is only necessairy to show ...
0
votes
0answers
23 views

What kind of information does the derivative of a sequence of functions give us?

When we derivate a function we know for example when then function increases and drecreases but when I derivate a sequence of functions I don't how to interpret the derivative since it is a collection ...
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2answers
36 views

Disc of convergence of a power series

Find the disc of convergence: $$\sum_{n=3}^\infty \left(1-\frac{1}{n^2}\right)^{-n^3}z^n$$ I have been manipulating the power series and I am pretty sure it has something to do with $e$ but I cannot ...
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2answers
45 views

What is the value of integral? [on hold]

Let $y(t)$ be a continuous function on $[0,\infty)$. If $$ y(t)= t\left(1-4 \int^t_0 y(x) dx\right) +4 \int^t_0 xy(x) dx$$ then what is the value of $\int^{\frac{\pi}{2}}_0 y(t) dt\,$?
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1answer
42 views

Decomposition of function of bounded variation

Suppose we have $f:\mathbb{R} \rightarrow \mathbb{R}$ which is of bounded variation. I would like to show that it can be presented as a sum of left and right continuous functions of bounded variation. ...
0
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2answers
24 views

Show if one series converges absolutely then so too does the other.

Task at hand: Let $a_n$ and $b_n$ be nonzero complex numbers for $n=1,2,3...$ . Suppose $\lim_{n\to \infty} \left|\frac{a_n}{b_n}\right|=l$ exists, and $l\neq0,\infty.$ show that if one of the series ...
1
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4answers
37 views

Pointwise convergence to zero, with integrals converging to a nonzero value

For $n\in{\mathbb{N}}$ let $$f_n(x)=nx(1-x^2)^n\qquad(0\le x\le 1).$$ Show that $\{f_n\}_{n=1}^\infty$ converges pointwise to $0$ on $[0,1]$. Show that $\{\int_0^1f_n\}_{n=1}^\infty$ converges to ...
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2answers
25 views

Alternative version of definition of convergence

Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$. Consider a new ...
1
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1answer
78 views

Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$

Having $f(z)=\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n$ I had to find what $\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n\sum\limits_{n=0}^{+\infty} \frac{D_n}{n!}z^n=\sum\limits_{n=0}^{+\infty} ...
1
vote
1answer
16 views

Heine definition of limit of a function at infinity using sequences

I couldn't find the answer neither on Google, nor this website, so decided to ask. The Heine definition of limit: from Wikipedia $\lim_{x\to a}f(x)=L$ if and only if for all sequences $x_n$ (with ...
0
votes
1answer
72 views

der(der(A)) and der(A)

I have a question about cluster points that would like to ask you, this question is one of the exercises in my textbook. Question: If $A$ is any subset of $R^d$, then $der(der($A$))$ is the set of ...
1
vote
0answers
24 views

Showing that the inferior integrals of two functions are equal.

Let $g:[a,b]\to\mathbb{R}$ Riemann integrable and put $h(x)=g(x)$ if $x$ is rational and $h(x)=g(x)+1$ if $x$ is irrational. Show that the inferior (Darboux) integral of $h$ is equal to the integral ...
1
vote
3answers
63 views

Solve the following sequence problem

Let a sequence be defined as $$a_n=\lim_ {x \to 0}{1-\cos (x)\cos(2x).....\cos (nx)\over x^2}$$ a)prove that the given sequence is monotonic and that it is not bounded above. b)calculate $$\lim_{n \to ...