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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0answers
14 views

(uniformly continuous) is it true?

Suppose $f$ is continuous on $(0,\infty)$ and $f$ is simmilar $g$ for all $x>M$ ($M>0$). (i.e. far any $\epsilon >0$, there is $M>0$ such that if $x>M$, then $|f-g|< \epsilon$) is ...
3
votes
0answers
25 views

Is $f(x)$ constant under these conditions?

Statement Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be an function that is concave up and increasing. If $\displaystyle \lim_{x\to \infty}\frac{f(x)}{x}=0$, then $f$ is constant. It'll be easy if ...
-1
votes
1answer
13 views

Let $D'$ be the set of all accumulation points of $D\subset \mathbb{R}$. Show that $D\cup D'$ is closed.

I am trying to prove: Let D ⊂ R and D′ the set of all accumulation points of D. Let D = D ∪ D′. Show that D is closed. I am very confused and unsure of how to do this. I would appreciate it if ...
2
votes
0answers
12 views

$\ell^p$-spaces for $p<1$

It is well known that whenever $p\in (0,1)$, the mapping $$ d_p(x,y):=\|x-y\|_{\ell^p}:=\left(\sum_{n=1}^\infty |x_n-y_n|^p\right)^\frac{1}{p} $$ turns $$\ell^p:=\{(x_n)_{n\in \mathbb ...
0
votes
1answer
15 views

Finding the mean of $x \mapsto |x|^p$ on the unit ball in $\mathbb{R}^n$ without calculating the volume of the unit ball.

Find the mean of the function $x\mapsto |x|^p$ on the ball $\{x:|x|<1\}\subset\mathbb{R}^n$ for $p\in(0, \infty)$. (Hint: You do not need the volume of the ball). I tried doing something ...
0
votes
0answers
11 views

Low-rank approximation proof

I am reading Low-rank approximation proof (theorem 5.8) in Trefethen & Bau book. It wrote: A is an $m \times n$ Matrix. For every $v$ with $0 \leqslant v \leqslant r$, define $$ ...
-1
votes
2answers
27 views

Why does the norm of a linear functional $T$ satisfy $\|T\|_*=\sup\{|T(f)|\mid f \in X, \|f\|\leq 1\}$?

For a normed linear space $X$, a linear functional on $X$ is said to be bounded provided there is an $M \geq 0$ for which $|T(f)|\leq M\|f\|$ for all $f \in X$. Denote $\|T\|_*$ the infinmum of all ...
3
votes
0answers
18 views

Is it natural how $L^p$ spaces measure local and global sizes the same?

This is a continuation of my question Spaces of functions similar to $L^p$ but with different local and global sizes. I have been bothered by the fact that the $L^p$ norm on $\mathbb R^n$, which is ...
1
vote
2answers
15 views

Is this proof correctly written? Show that the sum of two uniformly continuous functions on $A$ is uniformly continuous on $A$

If $f$ and $g$ are uniformly continuous functions in $A$ show that $f+g$ is uniformly continuous in $A$. Proof: because $f$ and $g$ are uniformly continuous on $A$ we can write ...
0
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0answers
19 views

Is it possible to find $g(\kappa)$ in this equation

I have ran into the following integral equation as part of my research. For $\xi = (\alpha\theta)^{1/\alpha}$ and for all $\theta>0$. I have the following equality $$\int\limits_0^\infty ...
3
votes
0answers
31 views

Differential equation exercise.

I am tasked with solving \begin{cases} y''(t) &=& \frac{(y(t)')^2}{y} - 2\frac{y'(t)}{y^4(t)} \\ y(0) &=& -1 \\ y'(0) &=& -2 \end{cases} I proceed by setting $v(s) = ...
0
votes
1answer
15 views

What happens when there is uniform convergence on an unspecified set?

Let $\{f_n\}$ define a sequence functions between metric spaces. I know what it means to say that "$f_n$ converges uniformly on some set $U$.". However, what if no set is specified, what does the ...
9
votes
5answers
485 views

when product of irrational numbers = rational number?

let $a$ and $b$ be irrational numbers. when do we have $ a \cdot b $ = rational number? for example $\sqrt{2} \cdot \sqrt{2}$=2. I was wondering if there some conditions for the product to be a ...
3
votes
2answers
39 views

$\epsilon $ $\delta$ proof of geometric series $x$<1

Give an $\epsilon -\delta $ proof of when $x<1$ $$ \sum ^ \infty _{k=0} x^k= \frac{1}{1-x} $$ Its been a more than a year from my analysis class. Trying to recall def of convergence of a ...
2
votes
4answers
34 views

Denseness of set S in R [duplicate]

Consider the set $S= \{\frac{p}{2^n} : p,n \in \mathbb{Z}\}$. Is it dense in $\mathbb{R}$ ?? To me intuitively it seems to be dense but I cannot come up with any analytic proof or disprove .
1
vote
0answers
21 views

Increase of Cantor's devil staircase functon

Cantor's function has an arithmetical presentation given as follows. Express $x$ in base 3. That is for $$ x \mapsto \sum\limits_{k=1}^{N} \frac{a_k}{3^k}, \ \ a_k \in \{ 0, 1, 2 \} $$ ...
2
votes
5answers
87 views

Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?

Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$? I get the feeling it isn't compact, but I can't figure out a way to prove this. I understand that, in $\Bbb Q$, any open set $U = \Bbb Q \cap ...
1
vote
0answers
10 views

If $u_{k|_{\Omega}} \to u$ in $W^{1,p}(\Omega)$ with $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ then $u_k \to u $ in $W^{1,p}(\bar{\Omega})$

I have already proven that for every $u \in W^{1,p}(\Omega)$ with $1 \leq p < \infty$ and every open set of class $C^1$ there exists a sequence $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ such that ...
1
vote
2answers
19 views

What is the definition of $f \in C^{\infty}(\overline{U})$?

I am reading Evans' Partial Differential Equations, Chapter $5$. I am concerned about the notation $f \in C^{\infty}(\overline{U})$, for an open set $U$. Does it mean $f$ is $C^{\infty}(U)$ and that ...
0
votes
0answers
15 views

Convergence of a mixed distribution

Let $Y_n=Z+\sum_{i=1}^n \delta_{1/i^2}$, with $\delta$ a point mass and $L(Z)=N(0,\sigma^2)$. Show that $\lim_{n\to\infty} Y_n=Y$, where $L(Y)=N(\frac{\pi^2}{6},\sigma^2)$ The answer file uses ...
-1
votes
2answers
36 views

Prove the function is continious.

If the function $f(x)$ is continious at $x=0$, using definitions show that $f(rx)$ is continious at $x=0$. Here $r$ is a real number.
1
vote
1answer
40 views

Derivatives of the Dirac delta function

From what I understand the Dirac's Delta derivatives have the meaning $$\int_{-\infty}^{\infty}\delta^{(k)}(x)\phi(x)dx=(-1)^k\int_{-\infty}^{\infty}\delta(x)\phi^{(k)}(x)dx$$ Assuming, of course that ...
1
vote
3answers
39 views

Parametrization of the sphere and the torus.

Is there a way to find easily the parametrization of the sphere and the tore ? I see on wikipedia that for the sphere it's $(x,y,z)=(\sin \theta\cos \varphi,\sin\theta\sin\varphi,\cos\varphi)$ with ...
2
votes
1answer
19 views

Showing Intermediate Value property and closed preimage implies continuity

Let $f : [0,1] \to \mathbb{R}$ be a function satisfying the Intermediate Value property. Assume that for any $y \in \mathbb{R},$ the preimage $f^{-1}(\{y\})$ is closed. Prove $f$ is continuous. ...
0
votes
1answer
52 views

Find a bijection between the Reals and an interval

I need to find a bijection between the reals and (−∞, 0) however I'm struggling to do so. I'm trying to prove that these two have the same cardinality. Also need to prove that the Reals and the ...
0
votes
1answer
54 views

A term for a function $f$ such that $x f'(x)$ is decreasing

Consider a differentiable, monotonically increasing function $f$. If $f'(x)$ is increasing, then $f$ is convex. If $f'(x)$ is decreasing, then $f$ is concave. Is there a term that describes $f$ when ...
0
votes
1answer
24 views

Let $f_n\in\mathcal C(0,1)$, $f_n\xrightarrow{\mathrm{unif}}f$ on every compact $K\subseteq(0,1)$. Is $f$ uniformly continuous on $(0,1)$?

This is part of a question on an old preliminary exam in Analysis at my institution. I think my answer is sufficient but I am not confident about it, and would appreciate feedback.
2
votes
0answers
37 views

$\exp(x)$ as defined by a net

Motivation: So, I had an idle thought last week, and I thought I would ask it here before I forget about it. It is well known that we can define $$ e^x = \lim_{n \to \infty} \left(1 + \frac x{n} ...
4
votes
1answer
53 views

Saturation of a measure Folland Problem 1.3.16

Exercise 16 - Let $(X,M,\mu)$ be a measure space. A set $E\subset X$ is called locally measurable if $E\cap A\in M$ for all $A\in M$ such that $\mu(A) < \infty$. Let $\tilde{M}$ be the ...
0
votes
1answer
22 views

Assume f and g are defined on all of $\mathbb{R}$ and that $\lim_{x\to p} f(x) = q $ and $\lim_{x\to q} g(x) = r $.

(a) Give an example to show that it may not be true that $\lim_{x\to p} g(f(x)) = r$ If we are to assume that f and g are defined on all of $\mathbb{R}$, wouldn't that mean that f and g are ...
0
votes
0answers
20 views

Result of a decay condition

Assuming that a function g is such that $ g(x) \leq C ( 1 + |x|)^{(-1 - \varepsilon)}$ for some $\varepsilon > 0$ , then how can we prove that $ \sum_{n = - \infty}^{n = + \infty} | g(x- k - ...
0
votes
2answers
47 views

Sequence and convergence(limit): $\lim_{n\to\infty} \frac{2^n}{3^{n+1}}$

Someone please help me with this problem with detailed explanation : Limit of $\frac{2^n}{3^{n+1}}$ as $n$ tends to $\infty$ ?
0
votes
1answer
30 views

is $[0,1]\backslash \mathbb{Q}$ totally bounded?

I learnt totally bounded by myself. Now, I am still trying to understand the definition and looking for counterexamples which is totally bounded but not compact. The below is some of counterexamples: ...
-2
votes
2answers
25 views

Double integration of Greatest integer function [on hold]

Integration of double integral of. $$ \int_0^2\int_0^{y-2} \lfloor x + y \rfloor \,dx\,dy $$
0
votes
2answers
19 views

Bound on oscillation of product of functions.

For $f,g:\mathbb{R}^n\to [-M,M]$, prove that $$\rm{Osc}_{fg}\leq M(\rm{Osc}_f+\rm{Osc}_g)$$ Where $$Osc_{f}(x_0)=\lim_{\varepsilon\to0}(\sup_{x\in B_\varepsilon(x_0)}f(x)-\inf_{x\in ...
-2
votes
1answer
45 views

Continuous in topology [on hold]

Let $f$ be a continuous function from the closed unit interval $[0,1]$ to itself. Show that there exists $t∈[0,1]$ such that $f(t)=t$. P.S Not use intermediate value theorem. Tomorrow this problem ...
1
vote
0answers
58 views

Compute the sum $\sum_{i=1}^{\infty}\frac{x^ {3i}}{(3i)!}$ [on hold]

Compute the sum $\sum_{i=1}^{\infty}\frac{x^ {3i}}{(3i)!}$. I have no idea to find this sum. Can anyone give me a hint? Thank you in advance !
3
votes
0answers
43 views

Proof that $\int_1^e\sum_{n=1}^\infty\frac{\sin(\ln nx)}{n^2+x}dx=\sum_{n=1}^\infty\int_1^e\frac{\sin(\ln nx)}{n^2+x}dx$

In a recent exam I took I was asked to prove that $$\int_1^e\sum_{n=1}^\infty\frac{\sin(\ln nx)}{n^2+x}dx=\sum_{n=1}^\infty\int_1^e\frac{\sin(\ln nx)}{n^2+x}dx$$ In my answer, I showed that for all ...
0
votes
1answer
24 views

Prove that if $g(x)$ is uniformly continuous in $(a,b]$ and $[b,c)$ then is uniformly continuous in $(a,c)$

I open this question to check my own proof and to ask a related question. My proof: if $g(x)$ is uniformly continuous in $(a,b]$ and $[b,c)$ then ...
5
votes
1answer
63 views

Spaces of functions similar to $L^p$ but with different local and global sizes

Obviously much of analysis on $\mathbb R^n$ considers $L^p$ spaces or other Banach spaces derived from them. The definition of $L^p(\mathbb R^n)$ looks very natural, but I've been bothered for some ...
2
votes
1answer
51 views

Does $\frac1n\sum\limits_{k=1}^na_k^2\to\rho$ with $0\le\rho<1$ imply$\prod\limits_{k=1}^na_k\to0$?

Let $\{a_n\}$ be a sequence of real numbers such that $\lim_{n\to \infty}\frac{\sum_{j=1}^n a_j^2}{n}=\rho$, and $0\le\rho<1$. The goal is to check whether the following is true $$\lim_{n\to ...
0
votes
0answers
12 views

How can the supremum of a set A of Dedekind cuts ever not be a member of set A?

I'm reading about Dedekind cuts in Rudin's Principles of Mathematical Analysis and it has led me to the following question: Let's say I have some set $A$ of Dedekind cuts. Specifically, $A = ...
3
votes
1answer
33 views

Class of the derivative of a bilinear map

This question is more conceptual than other thing. We know that if $f:U\subset \mathbb{R}^n \to \mathbb{R}^m$, where $U$ is a open subset of $\mathbb{R}^n$, is differentiable, then the derivative of ...
1
vote
3answers
70 views

Show that $\sup(\mathbb{Q} \cap (a,b)) = b$

Let $a<b$ be two real numbers. Show that $\sup(\mathbb{Q} \cap (a,b)) = b$ and $\inf(\mathbb{Q} \cap (a,b)) = a$. This intuitively makes sense. Since a sequence of rationals will infinitely ...
0
votes
1answer
54 views

Uniformly bound exists for a continuous function sequence in a neighbhorhood of a convergent point?

Assume that $$\lim_{n\to\infty}f_n(x_0) < \infty$$ and also that $\forall n\ge1$, $f_n(x)$ is continuous in a neighborhood of $x_0$, say $(x_0-\delta_1, x_0+\delta_1)$. Besides, for any fixed $x$ ...
5
votes
1answer
59 views

What are vector norms used for?

I'm currently working with a computer science problem that requires me to build vectors that can return their own norms. Based on Wolfram Alpha's description, I think I have an idea of how this is ...
0
votes
2answers
43 views

Unit ball of $L^1$, $L^\infty$ and $C(X)$ is not strictly convex

I need to show that the unit balls of $L^1(\mu)$, $L^\infty(\mu)$ and $C(X)$ are not strictly convex. I have already shown that if $1<p<\infty$ then the unit ball of $L^p(\mu)$ is strictly ...
1
vote
0answers
24 views

Is being real analytic at a point equivalent of matching the Taylor Series around that point?

Let $U \subset \mathbb{R}$ be an open interval and let $x_0 \in U$. Let $f: U \to \mathbb{R}$ be defined by a power series around $x_0$ with radius of convergence $R > 0$, $$f(x) = ...
0
votes
0answers
14 views

Showing any bounded sequence in Holder space $C^{1/2}$ has a convergent subsequence in Holder space $C^{1/3}.$

Prove that any bounded sequence in $C^{1/2}([0,1])$ admits a convergent subsequence in $C^{1/3}([0,1]),$ where we say that $f \in C^{\alpha}([0,1])$ if $f$ is Holder continuous of order $\alpha.$ The ...
1
vote
0answers
26 views

An analogue of the Cauchy formula for radius of convergence for power series with arbitrary (non-integer) exponents

By Cauchy formula, the radius of convergence of the series $\sum_{n=0}^{\infty}a_nr^{n}$ is $\rho=1/\limsup\limits_{n\rightarrow +\infty}\sqrt[n]{|a_n|}$. Let $\{\lambda_n\}_{n=0}^{\infty}$ be an ...