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, integration through the fundamental theorem of calculus.

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Boundedness and Cauchy Sequence

If I have a sequence {$a_n$} that has the property of $\lim(a_{n+1}-a_n)=0$, does that make it a Cauchy Sequence. I think it doesn't because $a_n = \sum_{k=1}^n \frac{1}{k}$ is a counter example. ...
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6 views

Continuity of $k^{th}$ order statistic?

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$ such that for a given $x \in \mathbb{R}^n,$ $f(x)$ is the $k^{th}$ highest co-ordinate of $x.$ For example if $n = 4, k =2$ and $x = [2,4,7,5]$ then ...
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20 views

Discuss the continuity of the given function

Let $f(x)$ be the function defined on the interval $(0,1)$ by $f(x) = \left\{ \begin{array}{l l} x& \quad \text{if $x$ is rational}\\ 1-x& \quad \text{otherwise} \end{array} \right.$ Discuss ...
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30 views

$a_1=3$ and $a_{n+1}=\frac{a_n}{2} + \frac{1}{a_n}$. Show that it monotonically decreases and find the limit.

What I've done so far: I have proved that this sequence is bounded below by 0, which is a very rough estimate. I know that the infimum is $\sqrt2$. Anyway, the question first asks me to prove that ...
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1answer
25 views

Does a differentiable function map open intervals to open intervals?

I know that preimages are open for open images, since differentiability implies continuity. I suspect there is a counterexample to the above though. This is not homework, just study.
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27 views

Analysis of Integral of a continuous function

one more question today I've been thinking on... Prove that if $f$ is continuous on $[a,b]$, $0<a<b$, then $\int_{a}^{b} {f(t) \over t} dt$ $=$ $\int_{a}^{s} {f(t) \over a} dt$ for some $s \in ...
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1answer
23 views

$\|f'(x)\|_{L^p} \le C \|f(x)\|_{L^p}^{1/2} \|f''(x)\|_{L^p}^{1/2}$ for smooth $f$ with compact support

I'm trying to prove the following Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function supported on $[a, b]$ where $-\infty < a < b < \infty$. $2 \le p < \infty$. Then $$ ...
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1answer
12 views

Uniform convergence for bounded multiplied functions

problem: If $\{f_n\}$ and $\{g_n\}$ converge uniformly on a set $E$, and there exist constants $M$ and $N$ such that $|f_n(x)| \le M$ and $|g_n(x)| \le N$ for all $n \in \mathbb{N}$ and all $x \in ...
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1answer
29 views

Simple sequence convergence question…

I know the standard solution to the problem that if a sequence $\{s_n\}$ converges, then it's arithmetic mean converges, and to the same limit. But how about this question?: Assertion: Let ...
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1answer
22 views

checking uniqueness of solution to differential equations

I have given the following differential equations: (a) $\dot u(t)=-\sin(t)u^2(t)$ with $u(0)=\frac14$ (b) $\dot u(t)=7-u(t)$ with $u(0)=0$ I want to know if (a) and (b) have an unique solution ...
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29 views

When a measure-zero set size is bigger then another set size [on hold]

Help me prove this theorem: Let $A \subseteq \mathbb{R}$ be a measure-zero set, and let $B\subseteq \mathbb{R}$ be a set. So, if $ |A| \geq |B| $ then $B$ is also a measure-zero set.
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1answer
26 views

Limit of $h_n(x)=x^{1+\frac{1}{2n-1}}$

$\lim_{n\to\infty}h_n(x) = x\lim_{n\to\infty}x^{\frac{1}{2n-1}}$ where $h_n(x)=x^{1+\frac{1}{2n-1}}$. I understand that $\lim_{n\to\infty}x^{\frac{1}{2n-1}}$ goes to one but what I don't understand is ...
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4answers
35 views

$s_1 = 1$ and $s_{n+1}=(\frac{n}{n+1})s_n^2$ monotonically decreases?

Hi I came across this question on page 65 of Elementary Analysis by Kenneth A.Ross. I am given that $s_1 = 1$ and I need to show that $s_{n+1} = (\frac{n}{n+1}) s_n^2$ monotonically decreases. I'm ...
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1answer
20 views

Function continuous Uryson's lemma?

when we proved Uryson's lemma we checked that the function $f:X \rightarrow [0,1]$, where $X$ is a $T_4$ space, i continuous by checking whether $f^{-1}([0,a))$ and $f^{-1}((b,1])$ are open. $f$ is ...
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7 views

Question on complex valued local martingales

So I was reading and found that the following was given as an example of a complex valued local martingale: $M_t = e^{\int_0^t f(\omega,s)dB_s - \frac 12\int_0^tf(\omega,s)^2ds}$ with $f(\omega,s) = ...
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24 views

Uses of the Heine-Cantor theorem?

Heine-Cantor Theorem: Let $[a,b]$ be a compact interval, $f:[a,b]\to R$ be a continuous function. Then $f$ is uniformly continuous, i.e. $\forall \epsilon>0 , \exists \delta>0$ such that ...
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31 views

Restriction of a continuous map is continuous.

First of all: sorry if it is the wrong tag. A map $f : A \to \mathbb{R}^m$ is continuous if there exists an open set $B \subset \mathbb{R}^n$ and a continuous function $G: B\to \mathbb{R}^m$ such ...
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1answer
16 views

Apply the Fourier Transform to $A\cdot e^{-a|k - k_0|}$

I have the following problem: The task is to show that $$f^*(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(k) e^{ik(x-vt)} dk$$ with $f(k) = A\cdot e^{-a|k - k_0|}$ equals $$f^*(k) = ...
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1answer
35 views

$f:\mathbb{R}\to\mathbb{R}$, $f(x)=x$ , $x$ rational and $1-x$, $x$ irational. Prove that $f$ is injective and isn't monotone on any interval [on hold]

For injective i have to take a string with rational numbers and a string with irational numbers. Please help me!
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1answer
16 views

there exists two sets $A,B$ such that $A \times B \subset E$ and $0<m_{1}(A)m_{1}(B)$ for $m_{2}(E)=1$

Suppose $E \in [0,2] \times [0,2]$ and $m_{2}(E)=1$ where $m_{2}$ is the two-dimensional Lebesgue measure. Show that there exists two sets $A,B$ such that $A \times B \subset E$ and ...
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1answer
28 views

How would this be proved using continuity?

k is a real continuous function such that $k(a)< 0$ for some a and k(x) tends to 0 as x tends to positive and negative infinity. Prove that k is bounded below and attains its lower bound.
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1answer
14 views

negative cauchy function

attempt: from defn of uniformally continuous $\forall \epsilon > 0 $ $\exists \delta > 0 $ s.t. $|x-y| < \delta $ with $x,y \in (-\infty,0)$ $\implies |f(x) - f(y)| < \epsilon$. My idea ...
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1answer
25 views

Uniform Convergent question. Having trouble dealing with the $x^n$ term.

How would you show that $(1-x) x^n$ is uniformly convergent on $[0,1]$?
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3answers
31 views

Question involving the Mean Value Theorem

Say you have a real function $h$ that is differentiable at every point and that $h(0)=0$ and $h(2)=0$ and $|h'(x)|\leq 1$ for all $x$. I know how the function looks like but how do you use the mean ...
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2answers
21 views

Uses of Jacobian of a map on $\mathbb{R}^n$.

For a map $f:\mathbb{R}^n\to\mathbb{R}^n$, Jacobian matrix of $f$ is defined as $$\begin{bmatrix} \frac{\partial f_1}{x_1}& \frac{\partial f_1}{\partial x_2}& \ldots \frac{\partial ...
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28 views

Tangent even derivative

How to show that $\tan^{(2n)}(0)=0$ for each $n \in \mathbb{N}$ ? I can't quite see the inductive step (or maybe there's another way to do it).
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11 views

smoothnes of rotationally symmetric functions

Consider a smooth function $f : [0,\infty[ \to \mathbb R$. This function induces a rotationally symmetric function $F : \mathbb R^2 \to \mathbb R$ via $F(x,y) = f(||x,y||)$. It should be correct ...
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17 views

How to prove this by inverse function theorem?

Assume h(t)=arctan(t)-t/(1+$t^2$/3) prove that $0<arctan(y)< pi/4$ implies arctan(y) is greater than y/(1+($y^2$/3))
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If f1 and f2 are 2 continous functions then f = { f1(x), x is rational and f2(x) , x is irational is continous in x0 if only if f1(x0)=f2(x0)

I have to prove that. I know that i have to take a string with rational number and a string with irational numbers but i don;t know how to do next. Please help me !
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3answers
26 views

$(1+\frac{a_n}{n})^n \to e^x$ if $a_n \to x$

I know that $\displaystyle \lim_{n\to \infty}{(1+x/n)^n}=e^x$ for every $x\in \mathbb R$. I want to prove that if $a_n\in \mathbb R$ is a sequence such that $a_n\to x$ then $(1+\frac{a_n}{n})^n \to ...
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1answer
24 views

Minimum of sum of increasing and decreasing function

Suppose we have a function $f(x)$ defined for integer $x$ in some bounded interval, which is positive and increasing $$f(x+1)\geq f(x)\\ f(x)>0$$ , and a function g(x) which is positive and ...
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1answer
29 views

Convergence of $\int_{1}^{\infty}\frac{t^q}{1+t^q}dt; q\in{\mathbb{R}}$

How do i proceed to check the convergence\absolute convergence of the integral $$\int_{1}^{\infty}\frac{t^q}{1+t^q}dt; q\in{\mathbb{R}}.$$ Can anyone help me with this please.
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1answer
19 views

Partial derivatives of polynomial in two variables

Let $k \in \mathbb N$, $a_{ij} \in \mathbb R$ for $i,j \in \mathbb N$, $i+j \le k$. A function $f:\mathbb R^2 \to \mathbb R$ $$f(x,y) = \sum_{i+j \le k} a_{ij}x^iy^j$$ is called polynomial of degree ...
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2answers
32 views

Recursive formula for definite integral

The integral is: $$I_n = \int_0^{\pi/4} \tan^{2n}x\,dx$$ I'm supposed to find the recursive formula.
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10 views

fourier series of $f \circ R_\alpha$

I have a problem with the following: If $f \in L^2 (T)$ and $\displaystyle f(x)= \sum_{n=-\infty}^{\infty} a_n e^{2 \pi i nx } $ in $L^2(T)$ the Fourier expansion, then why the Fourier expand of $ f ...
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1answer
36 views

$\int_{-1}^{1} x^{k+i} P_n(x)dx$, $P_n$ Legendre polynomial.

I was wondering whether there is a way to say what $$\int_{-1}^{1} x^{k} P_n(x)dx$$ is, where $k,n$ are positive integers or zero and $P_n$ is the n-th Legendre polynomial? I am looking for an ...
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1answer
24 views

Uniform Continuity and Boundedness Proof

Does uniform continuity imply boundedness? I know this question has been asked many times on this site. However, I found different answers from different people. Uniform continuity and boundedness ...
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64 views

How to prove $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ converges?

Could you please give me some hint how to prove this statement: If $0<a_n<1$ for each n, then $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ ...
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34 views

Proving that $X$ is a closed subset if and only if whenever $B_\epsilon (x) \cap X \neq \emptyset $ for every $\epsilon >0$, then $x \in X$.

Suppose $X$ is a subset of a metric space $M$. I would like to prove that $X$ is a closed subset of $M$ if and only if whenever $x$ is a point in $M$ such that $B_\epsilon (x) \cap X \neq \emptyset $ ...
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49 views

Show that $f(x,y)= \|x-y\|_2^2$ is differentiable

Problem: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$Note: $\| \cdot ...
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2answers
21 views

Integrable functions and absolute values

I have qutoted that the absolute value of an integral is less than or equal to the integral of an absolute value of a function. I have also said $|-g(x)| \le g(x) \le |g(x)|$ implies the integral ...
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16 views

supremums & functions

I'm am having some trouble with the below and was wondering if anyone would be kind enough to help me out. I have the following: $|L^n_{j}|\leq |\frac{\Delta t}{2} u_{tt}| + |\frac{ah}{2} u_{xx}|$ ...
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1answer
19 views

Cauchy sequence and uniform continuity

I read somewhere that because uniform continuous function maps cauchy sequence to cauchy sequence and cauchy sequence is bounded, so the function must be bounded. I am not sure if it is correct. My ...
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2answers
18 views

Limit of Rational Expression with Repeating Integer

I'm wondering if someone could point me in the right direction for the following problem: I'm trying to express $(a + aa + \cdots + a\cdots a)/10^n$ as a summation that I can take the limit of as ...
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2answers
31 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
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1answer
42 views

If f and g are monotone functions, such that f is continuous and f(x)=g(x) for rationals x, then g is also continuous

Let f and g be monotone functions on R such that f is continuous and g(x) = f(x) for all rational numbers x, then g is also continuous on R. My idea is a such: Without loss of generality, we assume ...
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0answers
34 views

The Implicit Function Theorem

In the implicit function theorem we have $f(x,y):\mathbb{R}^m \times \mathbb{R}^n\to \mathbb{R}^n$ and $a,x\in \mathbb{R}^m, \ b,y \in \mathbb{R}^n$ such that the $n\times n$ matrix ...
2
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0answers
15 views

Properties of the distribution function.

Let $\Omega\subset\mathbb{R}^n$ be open. Let $f:\Omega\rightarrow\mathbb{R}$ be a measurable function and $g\in L^p(\Omega)$, for some $p\geq 1$. For a measurable function $f: ...
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1answer
20 views

Showing the boundary of set has measure zero

Suppose $$ D = \{ (x,y,z) \in \mathbb{R}^3: 0 \leq x \leq 1 , \; x^2 \leq y \leq x, \; \; 0 \leq z \leq x \} $$ I want to show $\partial D $ has measure zero. My approach: I know that $D$ lies ...
2
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2answers
40 views

Uniform Continuity Help

Let $f$ be a continuous function on $\mathbb R$ such that $\lim_{x\to 0} f(\frac{1}{x})$ exists. Show that $f$ is uniformly continuous on $\mathbb R$. My proof is as follows: Let $n = \frac{1}{x}$ ...