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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Prove that g is differentiable

Question: Suppose f, g, and h are defined on (a,b) and $a < x_0 < b$. Assume f and h are differentiable at $x_0$, $f(x_0) = h(x_0)$, and $f(x) \le g(x) \le h(x)$ for all x in (a,b). Prove that g ...
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0answers
20 views

Proof of taking derivatives of both equal sides

I am curious about the proof of the following or whether the statement is true in general Assume that I have the following property: $f(x,y)=g(x,y,z)$ Can I assert that $D_xf=D_xg$ at any point ...
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1answer
23 views

Uniform convergence

Determine whether or not the given se ries of functions converges uniformly on the indicated interval (set) $\sum_{n=1}^\infty\frac{(1)}{(nx)^2}$ where x $\in (0,1]$ I don't know if we can apply can ...
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2answers
14 views

Real convergent sequences

Let $(a_n)$ be a bounded sequence for all $n$ such that $ \displaystyle a_n \geq \frac{1}{2} (a_{n-1}+a_{n+1})$ for $n\geq 2$. Show that $(a_n)$ converges. I think I cannot use any convergence tests ...
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0answers
12 views

A question about the definition of Lipschitz continuity

Suppose $f(x,y)$ is a function on $R^2$. If f is Lipschitz continuous with respect to y, then $|f(x,y_1)-f(x,y_2)|<C|y_1-y_2|$ for some constant C. But can anyone tell me that the constant C is ...
0
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1answer
10 views

invertibility, derivative, and difference quotient

Suppose that $f$ is an invertible differentiable function, that the domain of $f^{-1}$ contains an interval around $a$, and that $f^{-1}$ is continuous at $a$ and that $f^{-1}$ is continuous at $a$. ...
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11 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by definition it ...
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1answer
12 views

Sequence Lemma explanation

Then every neighbourhood $U$ of $x$ contains a point of $A$. So I don't see it happening unless $X$ is a metric space, but the proof is for any topological space.
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2answers
19 views

If $f$ is continuous at $x_0$ and $f(x_0)>M$, then $f(x)>M$ in some neighborhood of $x_0$

If $f$ is continuous at $x_0$ and $f(x_0)>M$, then $f(x)>M$ for all $x$ some neighborhood of $x_0$. My attempt is below. From the assumptions above, we have that $f(x_0) > M = f(x_1)$ for ...
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0answers
16 views

Continuous mapping problem

I have always confused on various "continuous mapping" problem. So here it is: Let $f:X_1 \rightarrow X_2$, $f$ is continuous. Then: if $X_1$ is open, is $X_2$ open? Similarly, if $X_1$ closed, is ...
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1answer
18 views

Question about monotonic subsequence theorem proof

This is a homework problem. Show that any sequence in $\mathbb{R}$ has a monotonic subsequence. Hint: start by supposing that is does not have a monotonically increasing subsequence. Here is my ...
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1answer
10 views

Non-singular derivative definition

I have a basic definition question. I am studying inverse function theorem, and I am stuck with what it means to say that for a $f'$ is non-singular? I looked it up in the internet, but it did not ...
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37 views

$\int\limits_{a}^{b} f(x) \operatorname d\!x = b \cdot f(b) - a \cdot f(a) - \int\limits_{f(a)}^{f(b)} f^{-1}(x) \operatorname d\!x$ proof

I just wanted to ask, if my proof is correct. I haven't seen the equation before, but I think it's quite useful. Let $f$ be an bijective differentiable function. Then the inverse function $f^{-1}$ ...
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1answer
21 views

$f, g: X \rightarrow \bar{R}$ are measurable, if $f \leq g$ a.e. then $\int f d\mu \leq \int g d\mu$

Let $(X,M,\mu)$ be a measure space. $f, g: X \rightarrow \bar{R}$ are measurable. If $f \leq g$ a.e. and $\int f d\mu, \int g d\mu$ both exist, show that $\int f d\mu \leq \int g d\mu$. Here a.e. ...
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0answers
13 views

Proving a Sobolev-Type inequality (also it is related to variational problem)

This is question 8.23 part $4$ from H. Brezis Functional analysis I already have that for any $f\in L^p(I)$, $p>1$ and $I=(0,1)$ there exists a unique $u\in H_0^1(I)$ satisfying ...
2
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2answers
24 views

Convergence of sequence of integrals.

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space, $f_n: \mathcal{X} \to \Bbb R$ a sequence of measurable functions, and $g_n:\mathcal{X} \to \Bbb R$ integrable functions such that $|f_n| ...
5
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6answers
59 views

$\int_{0}^{\infty} x \cdot \cos(x^3) dx$ convergence

$$\int_{0}^{\infty} x \cdot \cos(x^3) dx$$ I only want to prove, that this integral converges, I don't need to calculate the exact value. I don't know what to do with the cosinus, I can't get rid of ...
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0answers
24 views

A Question on Lebesgue Dominated Convergence Theorem

I have a general question about the dominated convergence theorem. The theorem states that if I have a sequence of measurable functions that are bounded by an integrable function and converge ...
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1answer
25 views

Calculating a limit of integrals

I am having a problem with the following exercise: Show that for every bounded borelian function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$, $\underset{n}{lim} \frac{n}{\sqrt{2\pi}} ...
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1answer
13 views

Getting $P^1$ by the sphere $S^1$

Consider $S^1=\left\{x\in\mathbb{R}^2: \lVert x\rVert=1\right\}$. Now let $P^1$ be obtained from $S^1$ by identifying antipodal point. I have to questions three this construction: 1) How can I ...
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0answers
16 views

Unbounded Function

I am trying to find vales of $a >0$ s.t the function is unbounded on $[0,1]$ $f_a(x)= \begin{cases} x^{a-2}(ax\sin(1/x)-\cos(1/x)), & x\neq 0 \\ 0, & x =0 \end{cases}$
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1answer
8 views

How to interpret this variance

If I have a probability measure defined by $P( \Omega ) = \int_{\Omega} (1-a) \delta(x) + a \delta(x-a^2) dx,$ then I noticed that the variance is given by $a^5(1-a)$. This is somewhat strange, cause ...
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1answer
26 views

complicated bijective differentiable function with surprisingly simple inverse function [on hold]

Do you know any complicated bijective differentiable functions with surprisingly simple inverse functions?
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2answers
14 views

What is $1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$?

I want to express $$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$$ in terms of standard functions (exp, cos, sin, etc.), but I just don't see what this function is. Does anybody here have an ...
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1answer
26 views

Proving that 0 is the greatest lower bound for this sequence

I'm ultimately trying to show a sequence converges to $0$, and I'm doing so by comparing it to another sequence, $n|c|^n$ where $|c|<1.$ I've already shown it's bounded and monotone decreasing past ...
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1answer
43 views

How to prove this sequence is convergent?

Suppose that the series $\sum_{k=1}^\infty{a_k}$ converges. Prove that $$\lim_{nā†’\infty}\frac{1}{n}\sum_{k=1}^{n}ka_k=0$$ I tried to use the definition of convergence of $\sum a_k$ ...
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1answer
13 views

$C^n$ for its projections implies $C^n$ for the map.

Let $n$ be an integer. Suppose that we have two open intervals $I_1,I_2$ in $\mathbb R$ and a function $f:I_1\times I_2\to \mathbb R$ such that for each fixed $x\in I_1$ the map $f(x,\cdot):I_2\to ...
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0answers
15 views

Banach fixed-point theorem and Newton

I have to combine the Newton method and the Banach fixed-point theorem: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. Let $f: I \rightarrow ...
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0answers
16 views

Inequalities absolute value

How do I verify if $|x^{1/\alpha}-y^{1/\alpha}|\leq c|x-y|^{1/\alpha}$, where $c$ is a constant real and $x,y>0$? I believe this is true.
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1answer
19 views

Proving the limit of a product of functions is equal to the product of the limits

Given $\lim_{x\to a} f(x) = \ell$ and $\lim_{x\to a} g(x) = m$. Prove that $$\lim_{x\to a} fg(x) = \ell m$$ So we want to prove that given some $\varepsilon>0$, we can find a $\delta>0$ such ...
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2answers
40 views

Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ with $f(x,y) = xy$ and $M=f^{-1}({0})$. Show that: The set $M$ is not a submanifold.

Assignment: Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ with $f(x,y) = xy$ and $M=f^{-1}({0})$. Show that: The set $M$ is not a submanifold. I've been able to show that sets are submanifolds ...
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1answer
32 views

How to prove that $\sup(A\cup B)=\max\{\sup(A),\sup(B)\}$? [on hold]

Let $A$ and $B$ be two bounded, nonempty set of real numbers. Prove that $$\sup(A\cup B)=\max\{\sup(A),\sup(B)\}.$$
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1answer
19 views

Proving the pre-image of $[0,1]$ is sequentially compact under this continuous function?

I'm given that to prove that $f:\mathbb R^n\rightarrow \mathbb R$ is continuous and that $\forall u\in \mathbb R^n,$ $f(u)\geq \|u\|.$ I'm then supposed to show that $f^{-1}([0,1])$ is sequentially ...
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0answers
18 views

Fourier transform with the function in denominator

I'm trying to do a Fourier transform of this function $$\frac{\bigtriangleup f(\textbf{x})}{1+f(\textbf{x})}$$ in terms of $\mathcal{F}(f(\textbf{x}))$. (Just like here ...
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4answers
40 views

In proving “if a set is compact, then it must be closed”, why does the finite subcover behave differently than the infinite open cover?

Proof from http://en.wikipedia.org/wiki/Heine-Borel_theorem: '''If a set is compact, then it must be closed.''' Let $S$ be a subset of $\mathbb{R}^n$. Observe first the following: if $a$ is ...
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2answers
36 views

Prove this is not a Cauchy sequence

If $$ x_{n}:= \sqrt{n}$$ show that$$ x_{n}$$ satisfies $$ \lim_{n\to \infty}|x_{n+1}-x_{n}|=0 $$ but that it is not a Cauchy sequence.
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0answers
23 views

Does there exist such function?

Fix an integer value $k\geq 1$. Let $[0,1]$ the unit interval and let $s\in [0,1]$. Does there exist a function $f$ (which depends on $k$ of course but not on $s$) such that $$\int_s^1 \left( ...
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3answers
39 views

Using the triangle inequality

How would I go about proving that $$|a|+|b| \leq |a+b|+|a-b|$$ Using the triangle inequality? I tried squaring both sides, yielding: $$|a|^2 +|b|^2 +2|a||b| \leq 2|a|^2+2|b|^2+2||a|^2-|b|^2|$$ Is ...
2
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1answer
19 views

Improper integral: is it convergent?

Is this integral finite? $$\int_s^t \frac{dx}{x^{1/2} - s^{1/2}}$$ where $s,t \in (0,\infty)$. More generally, I have the following integral $$\int_s^t ...
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3answers
23 views

Supremum and infimum of set $A=\{\,x \in\mathbb R : (x - a) (x - b) (x - c) (x - d) < 0\,\}$., where $a < b < c < d$

$\sup\{\,x \in\mathbb R : (x - a) (x - b) (x - c) (x - d) < 0\,\}$, where $a < b < c < d$
2
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1answer
16 views

Which of the following are true about sequences?

If $(x_n)$ is a sequence of real numbers such that for every $n$ we have $0<x_n<\frac{1}{n}$ then which of the following is true? $1.\lim_{n\to\infty}x_n=0$ $2.$If $f$ is continuous function ...
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2answers
17 views

Which of the following function need not necessarily has limit?

If $f(x)$ is a function such that $\lim_{x\to a}f(x)=L$ exists,then which of the following function need not necessarily has limit? $1.|f(x)-f(a)|$ $2.\frac{\sin(f(x))}{f(x)}$ $3.log(f(x))$ ...
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0answers
4 views

Bounding a function pointwise from bound on the expectation

Suppose we have a Lipschitz continuous positive function with bounded expectation Eg < a. What can be said about the point-wise bound ?
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2answers
51 views

How to prove this is NOT a Cauchy sequence

Show directly (from the definition) that the following are not Cauchy sequence $$n +\frac{(-1)^n}{n}$$
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1answer
14 views

Equivalent Statement of Chain Rule

This is an elementary question. The following theorem is from Principles of Mathematical Analysis: Chain Rule Suppose that $E$ is an open set in $\mathbb{R}^n$, $f$ maps $E$ into $\mathbb{R}^m$, $f$ ...
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0answers
23 views

Self-adjoint operator and eigenbasis

Let us assume that we have a self-adjoint operator $A: D(A) \subset L^2 \rightarrow L^2$ and we know that $A$ has a purely discrete spectrum and the eigenvalues of $A$ are simple. Does that mean that ...
0
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0answers
23 views

Real analytic function in one point and zeros

Let $f:I\to\mathbb{R}$ be a continuous function and $t_0\in I$ such that $f$ is analytic at $t_0$ and $f(t_0)=0$. Is it true that there is an entire neighbourhood of $t_0$ in which $f$ has no other ...
4
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0answers
85 views

Integral Contest [on hold]

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
0
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0answers
12 views

f is continious and g is defined as the integral of f. Now how can I show that g'(1)=f(1)

Im trying to solve a question I had on my exam but Im not sure how I should go about solving it. I dont even know in what direction I should look if I want to solve it. I am hoping someone here has a ...
2
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0answers
20 views

Real Analysis Differential functions

I am currently working through an exercise set and I am a little stuck on the following question: For $a > 0$, define a function $f_a(x)= \begin{cases} x^a \sin(1/x), &x \ne 0\\0, &x=0 ...