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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31 views

Epsilon delta proofs of theorems of continuity

Can anyone suggest a book which contains epsilon delta prooves for properties and theorems of continuity rather than sequential proofs.
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1answer
29 views

Approximating a Riemann integrable function using a continuous function

Let $f$ be Riemann integrable on $[a,b]$. Show that for every $ε > 0$, there is a continuous function $g$ on $[a,b]$ such that $$\int_a^b |f(x)−g(x)|\mathrm dx < ε. $$
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1answer
30 views

Does this conjecture in 1-d real analysis seem reasonable

Hello I am currently trying to prove a result and I have basically whittled it down to showing the following is true. Let $I\subset\mathbb{R}$ be an interval and fix $\alpha>1$ real number. Fix ...
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1answer
39 views

Weak convergence and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ imply norm convergence.

Consider a $\sigma$-finite measure space $(X,A,\mu)$ and $f,f_n\in L^p(\mu)$ with $1<p<\infty$. If $f_n \stackrel{w}{\to} f$ and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ hold, then ...
2
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1answer
25 views

“Convergence”/“Divergence” of $\prod_{n=1}^\infty (1 - \gamma_n)$

While trying to understand a proof of a result in an article, I stumbled upon the product $$\prod_{n=1}^\infty (1 - \gamma_n)$$ with $\gamma_n$ a real scalar belonging to $(0,1)$. I'm not really a ...
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1answer
31 views

Use Rolles Theorem to show that the function $x^{n}+kx+l=0$ has at most 2 roots if $n$ is even?

"Use Rolles Theorem to show that the function $x^{n}+kx+l=0$ has at most 2 roots if $n$ is even, and at most 3 roots if $n$ is odd?" To do this, I assume I must show that there are certain values of ...
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0answers
15 views

Triangle inequailty for $L^p$ norm to power $p$

I would like to prove the sharp estimate for the $L_p$ norm to power $p$ with $1\leq p <\infty$. What is the constant $C$ here: $$\left\|\sum_{j=1}^Jf_j\right\|^p_p\leq C\sum_{j=1}^J\|f_j\|_p^p$$ ...
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89 views

Evaluate $\sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1}$

I am trying to re-learn some basic math and I realize I have forgotten most of it. Evaluate $$\sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1}$$ Call the terms $S_n$ and the total sum $S$. $$S_n < ...
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2answers
19 views

Property of real functions when derivative approaches zero

This is a question from my exam in Calculus 1. Problem 6 Let $f: [0,\infty[ \to \mathbb{R}$ be continuously differentiable and $\lim_{x \to \infty} f'(x) = 0$. a) Show that for $n \in ...
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51 views

Continuity and differentiability for $\sin(\sqrt x)$ & $\sinh(\sqrt {-x})$?

Let $f: \Bbb R \to \Bbb R$ with $$f(x)= \begin{cases} {\sin(\sqrt{x})\over\sqrt{x}},& \text{for } x>0\\ 1,& \text{for } x=0\\ ...
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0answers
23 views

Closed subset for $L^2$ strong and weak convergence

I was trying to solve the following exercise. Let $K$ a closed subset of $\mathbb{R}$. $$X:=\{f\in L^2[0,1]:f(x)\in K \:a.e.\:x\in [0,1] \}$$ Then: 1)X is closed under strong convergence in $L^2$. ...
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1answer
37 views

Is the sequence $(-1)^n$ eventually or frequently in the set {$1$}

This is what I know: A sequence $(a_{n})$ is eventually in a set $A \subseteq \mathbb{R} $ if there exists an $N \in \mathbb{N}$ such that $a_{n} \in A $ $\forall n \geq N$ A sequence $(a_{n})$ is ...
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1answer
35 views

Countable and uncountable sets and functions on them

A function $f:R→R$ is given. Whenever we choose real numbers $a < b$ and set {$f(x) : a < x < b$} has a biggest element, we call this element local maximum of function $f$. Prove that ...
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2answers
43 views

Uniqueness of log function with relaxed conditions?

Question If: $$f(a) + f(b) = f(ab)$$ $$ f(1) = 0 $$ $$ a<b \implies f(a) < f(b) \forall a,b \in N $$ where $N$ is the set of natural numbers. Prove or disprove $f$ must be the $\log$ ...
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0answers
18 views

Step 2 of Strichartz's Estimate Proof.

I am stuck with Step 2 of the Strichartz's estimate. My question is actually a continuation of a topic which has been raised some time ago and it could be seen here Technical question about Strichartz ...
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0answers
58 views

Find interior points, boundary points, cluster points, limit points and isolated points of a set

Determine the interior points, the boundary points, the cluster points, the limit points, and the isolated points of each of the following subsets of $\mathbb R^2$. Also, classify each of the sets as ...
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2answers
131 views

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$. If $f(0)=0$, find the maximum value of $f(5)$. $f'(x)=f(x)$ ...
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1answer
36 views

Taylor polynomial $P(n)$ of $f$ at $x_0$. [on hold]

True or False: if $P(n)$ is the $n^{th}$ Taylor polynomial for $f$ at a point $x_0$, then the first $n$ derivatives of $P(n)$ and $f$ are equal at $x_0$.
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1answer
24 views

Function whose $n$-th derivative at $x=0$ is $n^3$ / evaluating the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$

I have troubles proving that the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$ represents the function $f(x)=e^x(x^3+3x^2+x)$. My idea was using the identity theorem for power series and the ...
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2answers
48 views

If $\{a_n\}$ and $\{b_n\}$ are increasing, then $\{a_n b_n\}$ is increasing.

For two sequences $a_n$ and $b_n$, “If $\{a_n\}$ and $\{b_n\}$ are increasing, then $\{a_nb_n\}$ is increasing.” Show this is false, make the hypothesis on $\{b_n\}$ stronger, and prove the amended ...
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1answer
30 views

Prerequisites for Spivak's Calculus on Manifolds

Hello I am a sophomore and I would like to know if Michael's other book named 'Calculus' is enough preparation for the Manifolds, if not please do tell what else should I be reading? I am taking an ...
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1answer
27 views

Proving function is infinitely differentiable [on hold]

On some interval, there is a function $h(x)$ that is differentiable and continuous. Then, there is another function $c(x)$ that is smooth(infinitely differentiable). Given that $h'(x) = c(h(x))$, show ...
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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 ...
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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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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
49 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
32 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, ...
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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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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 ...
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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: ...
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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 ...
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7answers
203 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.
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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
23 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
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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 ...
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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
33 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 ...