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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9 views

Prove this series converges to a continuous function

My problem: Prove that the series $\sum\limits_{n=0}^\infty e^{n(\sin(nx)-2)}$ converges for all $x\in\mathbb{R}$ to a continuous function. By the root test it converges, but as far as the continuous ...
0
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1answer
11 views

Two decreasing, convex functions agreeing on a closed set

Fix a closed subset $B$ of $[0,\infty)$ and assume that $0\in B$. I am striving to construct two functions $f,g:[0,\infty)\to\mathbb R$ such that $f(0)=g(0)=1$; $f(x)\geq 0$ and $g(x)\geq0$ for each ...
1
vote
1answer
13 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
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0answers
13 views
0
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1answer
20 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} ...
0
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0answers
10 views

Prove $|M_{T \wedge n}| \le c + K$

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
0
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0answers
8 views

Clarifications on proof of Doob's Forward Convergence Theorem, warning related to it and proof of a corollary

From Williams' Probability with Martingales: $X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$ --> Is this supposed to be stronger than $\lim X_n$ does not exist because it's ...
2
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0answers
14 views

Strong Solutions to Nonlinear ODE by Contraction Mapping

Consider the $1$-d ODE $$-u_{xx}+u-\epsilon u^{2}=f, \tag{1}$$ where $f$ is a nice RHS, say $f\in\mathcal{S}(\mathbb{R})$, and $\epsilon>0$. By using the Bessel potential, one looks for solutions ...
0
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0answers
7 views

Space of Riesz transforms is closed

Let $B=\bigoplus_{j=0}^nL^1(\mathbb R^n)$ a Banach space with norm $\|(f_0,\ldots,f_n)\|=\|f_0\|_{L^1}+\cdots+\|f_n\|_{L^1}$. Define $$S=\{(f_0,f_1,\ldots,f_n):f_j=R_jf_0,\quad j=1,2,\ldots,n\}\subset ...
2
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2answers
41 views

The Cantor staircase function and related things

The Cantor staircase function https://en.wikipedia.org/wiki/Cantor_function has an interesting property: $\{x\colon f'(x)\neq 0\}$ is a nowheredense nullset. But it it differentiable almost ...
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0answers
13 views

A set A is infinite iff there exists a bijection from A to a proper subset of A

Below is the proof I have in my book for the first part of supposition. Let A be a set. First suppose A is infinite. By the proposition saying "Every infinite set contains an infinite subset that ...
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3answers
41 views

Finding a Taylor Series Representation: $f(x)= \frac{x}{(1+4x^2)^2}$

Find Taylor series representations for the following function. For precisely what values of $x$ is the series representation valid? $$f(x) = \frac{x}{(1+4x^2)^2}$$
2
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1answer
34 views

The set where a derivative vanishes is G-delta

If $f:I\to R$ ($I$ - interval) is differentiable, then $\{x\colon f'(x)=0\}$ is a $G_{\delta}$ set. The lecturer didn't prove this fact and I found no proof in my books. How it can be proven?
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0answers
24 views

find the gradient

Let $h:\mathbb{R}^n \rightarrow \mathbb{R}^n$ defined by $h(x) = \phi(|x|^2)x$, where $\phi$ is a real valued function that is differentiable. Find a formula for $\nabla h(x)$ (i.e the gradient of ...
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0answers
11 views

Limit of Recursive sequences [on hold]

Given $Z_1 > 0$ and $a > 0$ $Z_{n+1} = (a+Z_n)^{1/2}$ To show $Z_n$ is convergent and find its limit
1
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1answer
33 views

Do there exist bump functions with uniformly bounded derivatives? [duplicate]

Let us consider a bump function $\phi: \mathbb{R} \longrightarrow \mathbb{R}$, smooth, with compact support. The most common examples are built from the function $$ \psi(x) = \begin{cases} \exp ( ...
3
votes
1answer
35 views

Why is $R((X))$ defined as follows?

Let $R$ be a commutative ring. Then $R((X))$ is defined as the set of all $\sum_{n\geq N} a_n X_n$ where $N\in\mathbb{Z}$ and is called "The Formal Laurent series". But why? Why don't we consider ...
1
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0answers
14 views

Show f takes on maximum boundary for function

Suppose $\Omega$ is a bound set in $\mathbb{R}^2$ and $\bar\Omega$ its closure. Assume $f\in C^2(\Omega)\bigcap C^0(\bar\Omega)$. Moreover, assume $f$ satisfies the partial differential ...
4
votes
2answers
106 views

Uniform unboundedness of linear operators

Question: Suppose that $(T_k)_{k=1}^{\infty}$ is a sequence of invertible linear operators on $\mathbb{R}^n$. Suppose that $\forall x \in \mathbb{R}^{n}\setminus \{0\}$, we have $$\lim_{k\to\infty} ...
6
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3answers
187 views

Ratio test for sequences, the other direction

Suppose I have a real sequence $x_n\to 0$. Is it true that: $$ \left|\frac{x_{n+1}}{x_n}\right|\to r<1 $$ for some $r\in\mathbb{R}$? If not, is it true that: $$\exists ...
1
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1answer
41 views

Is it possible to find an uncountable number of disjoint open intervals in $R$?

Is it possible to find an uncountable number of disjoint open intervals in $R$? Several times I saw the sentence every open set in $\mathbb{R}$ can be expressed as a countable number of open ...
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0answers
69 views

Lack of continuity between undergraduate mathematics and “real” mathematics [on hold]

First, let me begin with some background. My mathematical knowledge is largely a product of self-study and though some are sure to be skeptical for that reason, I do consider myself at least competent ...
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2answers
25 views

Derivative of n x n Invertible Matrix

For an invertible $n$ x $n$ matrix $A$, define $f(A):=A^{-2}$. Calculate the derivative $D\space f(A)$. (i.e. give $D\space f(A)B$ for arbitrary $B$.) I'm not super sure how to go about this?
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24 views

Proof that the Cardinality of Borel Sets on $\mathbb R$ is $c$ without using the ordinals .

I'm trying to prove that cardinality of Borel sets is $c$ without using the concept of Ordinal number ! I know that the Cardinal of Borel sets are greater than $c$ because of every point in $\mathbb ...
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0answers
17 views

how can I prove that?

A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in R. Assume $f \in L^1( \Bbb R^1)$, and prove that there is a sequence $\{g_n\}$ of ...
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0answers
26 views

Do partial derivatives determine function uniquely?

When we talk about functions of one variable, we have less complicated situations compared to the case in functions of several variables. In the one variable case, for continuous functions ...
0
votes
1answer
43 views

If the integral of $|f_n|$ converges to zero, so does the integral of $f_ng$ for integrable $g$

Let's assume that the $f_n$ are integrable and all bounded with the same bound. also, that $\int|f_n|\to 0$ as $n\to\infty$ (a) Prove under these assumptions that $\int f_ng\to0$ for any $g\in ...
0
votes
1answer
29 views

On a Cauchy problem exercise.

I can't seem to find the trick to solve the following Cauchy problem: \begin{cases} y' = \alpha( 1 - y/ \beta) y \\ y(0) = y_o \end{cases} where $\alpha$ and $\beta$ are greater than zero. Anyone ...
-1
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1answer
15 views

Discrete measure and piecewise function

Hi guys, can anyone please help me with why we can introduce a sectionally constant function that has support $\lambda_i, i \in \mathbb{N}$. I do not understand why we can do the part I marked with ...
0
votes
2answers
31 views

Explicit Integrals and LimInf/LimSup

(a) Show that $f(t):=\int_0^\infty e^{-tx}\frac{sin \space x}{x}dx$ exists for $t>0$ and defines a differentiable function $f$. Calculate $f'(t)$ for $t>0$ and evaluate it explicitly. (b) Prove ...
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0answers
21 views

Showing a function is defined

(a) Defined $f$ by $f(y):=\int_0^\infty\frac{xy}{(x^4+y^4)^{3/4}}dx$. Prove $f(y)$ is defined (i.e integral exists) for every $y\in\mathbb{R}$. (b)Prove that actually $f(y)=c\space sign \space y$ for ...
0
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1answer
15 views

Proving functions are in neighborhoods and their partial derivatives

Let $g:\mathbb{R}_+\rightarrow\mathbb{R}$ is smooth, $g(1)=1$, and $g'(1)\ne \space \lambda$ (a)Prove that the set $S:=\{(x,y,z)\in \mathbb{R}_+^3 \vert \space x+g(x)+y+g(y)+z+g(z=6)\}$ is locally, ...
3
votes
2answers
27 views

Convergence of the series $\sum (a^{1/n}-1)^{\lambda}.$

The series $$\sum_{n=1}^{\infty} (a^{1/n}-1)^{\lambda}$$ converges for $1.\lambda\geq0$ $2.\lambda\geq1$ $3.\lambda>1$ $4.\lambda\leq1$ I am confused about the convergence of the series. No ...
2
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2answers
41 views

conditionally convergent but not absolutely convergent series

I'm stuck on the following exercise: Let $\sum_{n=0}^{\infty} a_n$ be a series of real numbers which is conditionally convergent, but not absolutely convergent. Define the sets ...
2
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0answers
24 views

Show that there exists $ \lambda \ge 0$ such that $v=\lambda u$

Let $\Omega \subset \mathbb{R}^n$ be open. Let $u,v \in L^1_\text{loc}(\Omega)$ with $u \ne 0$ a.e on a set of positive measure. Assume that $$\phi \in C_c^\infty(\Omega), \int u\phi > 0 \implies ...
1
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1answer
28 views

How do I solve the following differential equation (It's not seperable)?

I'm trying to Solve the following equation: find the solution $y:(-1,1) \rightarrow \mathbb{R}$ of $y'=\dfrac{y}{1-x^2}+x$? It is not separable and I have no other Tools to solve it.
0
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1answer
16 views

Holders Inequality: Suppose $\int_{0}^\infty x^{-2}|f|^5 dx < \infty$. Prove that $\lim_{t \to 0} t^{-\frac{6}{5}} \int_0^t f(x)dx = 0$

I discovered last night that I have an error in my proof to the following problem and I need help fixing it (or need a new solution) $$ \text{Suppose that} \int_{0}^\infty x^{-2}|f|^5 dx < \infty. ...
0
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0answers
15 views

Are any tools or techniques available to solve the “placement of safety points” problem?

Definition 0. Given a metric space $X$ and subsets $H$ and $S$ thereof, define: $$d(H,S) = \sup_{h \in H} \inf_{s \in S}d(h,s)$$ Here's some extremely dodgy intuition. Imagine $S$ is a set of ...
0
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3answers
28 views

One of the Heire-Borel lemmas states the following:

If a set is closed and bounded in $\mathbb{R}^{n}$, then it is compact. However, what about a more abstract metric space $(X,d)$? Let $(X,d)$ be a complete metric space with a subset $A$ closed and ...
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1answer
45 views

Set of rational numbers bounded between two irrationals is a closed set?

Consider the metric space $\mathbb{R}$ equipped with the standard distance metric. Let $S$ be a set of rational numbers in the open interval $(a,b)$ where $a$ and $b$ are irrational. Prove that $S$ is ...
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0answers
23 views

Find the value of :$\lim_{\Delta t \rightarrow 0^+} \frac{\epsilon}{\sqrt{16\pi D (\Delta t)^3}}e^{-\epsilon^2/(4D(\Delta t))}$

Is this limit evaluation correct? $$\lim_{\Delta t \rightarrow 0^+} \frac{\epsilon}{\sqrt{16\pi D (\Delta t)^3}}e^{-\epsilon^2/(4D(\Delta t))} = \lim_{\Delta t \rightarrow 0^+} ...
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0answers
7 views

Error bound for function limit with arbitrary $\Delta h$

Say we have some function $f\in C^1$. I would like to somehow bound the error that is made when the limit is approximated using some arbirary $h$, that is to find $E(x,h)$ such that: ...
2
votes
2answers
33 views

Differential equation $y'=(1+f^{2}(x))y(x)$

Consider the Differential equation $$y'=(1+f^{2}(x))y(x), y(0)=1,x\geq0$$ where $f$ is a bounded continuous function on $[0,\infty).$ Prove that the given ODE has unique solution $y$ and ...
0
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0answers
23 views

$C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$

Show that $C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$, where $\Omega$ is an open subset of $\mathbb{R^n}$. My try: Let ...
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3answers
51 views

How to evaluate $\lim _{x\to 0}\left(\frac{xe^x-2+2\cos x-x}{\left(\sin x\right)^2 \tan\left(2x\right)+xe^{-\frac{1}{\left|x\right|}}}\right)$?

I have a problem with this limit, I don't know what method to use. I have no idea how to compute it. Is it possible to compute this limit with the McLaurin expansion? Can you explain the method and ...
2
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0answers
28 views

Proving that $f$ and $g$ are identically $0$ on the entire domain

Q: Let $f$ and $g$ be two non-decreasing twice differentiable functions defined on an interval $(a,b)$ such that for each $x\in (a,b)$, $f''(x)=g(x)$ and $g''(x)=f(x)$. Suppose also that $f(x)g(x)$ is ...
2
votes
1answer
17 views

Proving function is Schwartz

I want to prove that $f(t)=e^{-t^2C\pi}$ is Schwartz. I tried computing derivatives and showing that for all $n,k\in \mathbb{N}_0$ $$\lim_{t\to \infty}t^{k}f^{(n)}(t)=0$$ but it gets messy pretty ...
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0answers
23 views

Limit of monotone decreasing function on generalised inverse.

Consider a right-continuous, monotone decreasing, non-negative function $\bar F(x)$ (its the tail of a probability distribution, but that doesn't matter). Now let \begin{equation} I_{n}=\{x : \bar ...
0
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0answers
26 views

If $A \subseteq \mathbb{R}^n$ is path-connected, with path-connected complement, is $A$ necessarily simply-connected?

Suppose we're given a subset of $\mathbb{R}^n$, call it $A$, that satisfies the following. $A$ is path-connected $A^c$ is path-connected Q0. Does it follow that $A$ is ...
2
votes
0answers
54 views

continuous linear functional on $l^{\infty}$ space

Let $l_{\infty}$ be the space of all bounded complex-valued sequences equipped with the supremum norm. Consider the natural standard basis $\{e_n\}_{n \in \mathbb{N}}$ of $l_{\infty}$. For any ...