For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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19 views

Is this collection of function uniformly equicontinuous? Hints on the proof.

Let $f_n(x)=\frac{1}{n}\cos(e^{nx})$ for $n\in\mathbb{N}$ be a sequence of functions for $x\in[0,1]$. Is it true that $\{f_n\}$ is a uniformly equicontinuous collection of functions? My attempt so ...
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1answer
32 views

Using induction on modified inequalities.

Here's the original problem: Prove by induction that $\left(\frac{1}{2}\right) \left(\frac{3}{4}\right) \cdots \left(\frac{2n-1}{2n} \right) \leq \frac{1}{\sqrt{n+1}}$ for all $n \in \mathbb{N}$. ...
3
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1answer
88 views

Proving that the limit of an integral of a series exists

The goal is to show that the following limit exists $$\lim_{T\to\infty} \frac{1}{T}\int_{-T}^T f(x)dx$$ where $$f(x)=\sum_{n=1}^\infty \frac{e^{ia_n x}}{n^2}$$ I already showed that $f$ is bounded ...
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2answers
39 views

Closure of Union contains Union of Closures

I'm teaching my self topology using a book I found. This is the second part of a 4 part question. First part is here. I'm trying to prove the following problem from a book I found: Let $X$ be a ...
3
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1answer
47 views

Function with countably many points of discontinuity

Aside from rigor, is this proof correct? Claim. Let $f$ be a function defined on $[0, 1]$ such that $\lim\limits_{y\to a} f(y)$ exists for all $a \in [0, 1]$. Then for any $\epsilon > 0$ there are ...
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0answers
28 views

Closure of Intersection is Subset of Intersection of Closures

I'm trying to prove the following problem from a book I found: Let $X$ be a topological space and let $\mathscr{A}$ be a collection of subset of $X$. Prove $\overline{ \bigcap \limits_{A \in \...
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0answers
99 views

Find the limit of the mediant sequence $\frac{p_{n+1}}{q_{n+1}}=\frac{ap_n+bk_n}{aq_n+bm_n},~\frac{k_{n+1}}{m_{n+1}}=\frac{cp_n+dk_n}{cq_n+dm_n}$

How to find the general formula for the limit of the sequence: $$\frac{p_{n+1}}{q_{n+1}}=\frac{ap_n+bk_n}{aq_n+bm_n}$$ $$\frac{k_{n+1}}{m_{n+1}}=\frac{cp_n+dk_n}{cq_n+dm_n}$$ $$a,b,c,d>0$$ $$...
5
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3answers
88 views

Factorial Proof by Induction Question? [duplicate]

$\text{Use the PMI to prove the following for all natural numbers n.}$ $ \frac{1}{2!} + \frac{2}{3!} + \cdot \cdot \cdot + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!} $ So for this question I get ...
3
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8answers
104 views

Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$

Prove: $$\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$$ for all x, y positive $$\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}-\sqrt{x}-\sqrt{y}\geq 0$$ $$\frac{x\sqrt{x}+y\sqrt{y}-x\sqrt{y}-...
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1answer
26 views

Closed disjoint sets in $\omega_1$ implies one of them is countable

Let $A, B \subseteq \omega_1$ be disjoint, closed sets. Show that one of A or B must be countable. Since $\omega_1$ is uncountable, at least one of A or B must be countable, I want to show that ...
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1answer
57 views

Need help in understanding some basic ordinal concepts

I'm trying to construct a proof that there is no homeomorphism from a subspace of $\omega_1$ to $\Bbb{Q}$ with the usual topology, but first I need to clear up some concepts. This is the definition ...
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1answer
55 views

Proof of indexed cartesian product

Could anyone verify if my proof is correct please? The question is as follows: For each $(i,j)\in I\times I$, let $C_{i,j}=A_i\times B_j$, and let $P=I\times I$. Prove $\bigcup_{p\in P}C_p=\...
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1answer
16 views

How to show a set is compact using sequential compactness definition?

Let $l^{\infty}$ be the vector space of all bounded sequences $x=(x_n)$ of real numbers with the norm $||x||=\sup_{n\in\mathbb{N}}|x_n|$ and $l^{\infty}$ is complete. I am trying to show that the set ...
0
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1answer
18 views

Given any $r \in \mathbb Q$, there is $m \in \mathbb Z$ such that $m \le r < m + 1$

Suppose the set $S$ contains all $n \in \mathbb Z$ such that $n > r$ for any $r \in \mathbb Q$. By Archimedes, there are some $m, n \in \mathbb Z$ such that $n > r > m$ for any $r \in \mathbb ...
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0answers
32 views

Prove: $\mid\sum_{i=1}^{n} x_i\mid\leq \sum_{i=1}^{n}\mid x_i\mid$ [duplicate]

$\mid\sum_{i=1}^{n} x_i\mid\leq \sum_{i=1}^{n}\mid x_i\mid$ If $n$ is even we will divide the sum into groups of $2$ $x$'s namely $\mid x+x \mid \leq \mid x\mid+\mid x \mid$ and will repeat the ...
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2answers
43 views

There is group a $S$ with $2n$ members $n$ of them are identical and $n$ of them are different, How many subsets are there?

I have the following question : There is a set $S$ with $2n$ members $n$ of them are identical and $n$ of them are different, How many different subsets are there for $S$ in size $n$. This is what I ...
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1answer
55 views

Proving that every set has a finite subset of ``smallest elements''.

Update: I would like to have this deleted but I'm behind a firewall which prevents from using most features -- including commenting. Reason for deleting: $c-x\in \Bbb N_0^n\setminus\{(0,0,\dots,0)\}$ ...
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1answer
68 views

Finite Field Question: Which of the followings are true?

I have the following True or False question that I am having trouble getting it correct. I've written down my thoughts on each choice. If anyone could verify my thoughts or tell me where I made a ...
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2answers
53 views

Proving that $3+11 +19 + \cdots + (8n-5)= 4n^2 -n$ by induction

$ 3+11 +19 + \cdot \cdot \cdot + (8n-5)= 4n^2 -n.$ I rewrote the problem in sigma notation. Would this change the answer? $$ \sum_{i=1}^n (8n-5) = 4n^2-n.$$ $\color{red}{Proof :}$ (i) $\;$ The ...
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2answers
56 views

No subspace of $R_{usual}$ is homeomorphic to $\omega_1$

Where $\omega_1$ is defined as a subset of an uncountable well-order W s.t $\omega_1 = \{ \alpha \in W: \ pred(\alpha)$ is countable} with the following properties: 1. $\omega_1$ is uncountable For ...
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1answer
48 views

Principle Mathematical Induction Product Notation Question?

$$ \prod_{i=1}^n \left( 2i-1 \right) = \left( \frac{(2n)!}{n!2^{n}} \right) $$ My question for this proof is that I am trying to understand product notation and how to factor this question to its ...
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2answers
30 views

Proof verification: $\sin(Arccos(x))$ is always positive, regardless of $x$.

$\sin(Arccos(x)) \implies \sin(Arcsin(\frac{\pi}{2}-x))$ and since $-1 \leq x \leq 1$ then we can have as the maximal and minimal values in the $Arcsin,$ $\frac{\pi}{2}-(-1)\approx2.57$ and $\frac{\pi}...
0
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0answers
40 views

derivative is bounded $\iff$ functions is lipstchitz

I need to prove the theorem that says that, if the derivative of a function is bounded, then the function itself is lipstchitz continuous. I'm concerned about the $\leftarrow$ part of my proof. $\...
2
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2answers
36 views

Proving set of vectors is linearly independent

The question asks: If $\{v_{1}, v_{2}, \cdots, v_{n}, v_{n+1}\}$ is linearly independent, prove $ \{v_{1}, v_{2},\cdots, v_{n}\}$ is linearly independent. My attempt at a proof: If $\{v_{1}, v_{...
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1answer
22 views

Why is $V_{\frac{1}{2}} = \{ (\frac{1}{2}, y) \in X : y \in [0,1] \}$ not open in the lexicographical topology on unit square

Where X is the unit square with the lexicographical topology. I think its because taking any open set around $V_{\frac{1}{2}}$ say $(a \times b, c\times d)$ we can always find some x s.t $a < x &...
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2answers
44 views

I don't understand a little section of the Fundamental theorem of arithmetic.

I understand the theorem from a general point of view, but there's this little part, which I don't. This is the theorem, as explained by Richard Courant and Herbert Robbins: What I don't ...
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1answer
38 views

No One to One $C^1$ mapping of $\mathbb{R}^2\to\mathbb{R}$

My thought is to use the implicit function theorem. First, if $f$ is constant, then it is clearly not one to one. Thus $\exists p=(x_0,y_0)\in\mathbb{R}^2$ such that $\frac{\partial f}{\partial x}(p)\...
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0answers
43 views

Looking for feedback on proof (math logic)

$\def\sD{\mathscr{D}} \def\sC{\mathscr{C}} \def\sB{\mathscr{B}} \def\Gm{\Gamma}$ This is a theorem in Mendelson's Intro. to Math. Logic (pg 66, Proposition 2.4). I try to follow his conventions. MP ...
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1answer
40 views

Property singular support of the convolution of distributions

Let $u \in \mathcal{D}'(\Omega)$ and $U \subset \Omega$ open. By definition we say that $u \in \mathcal{E}(U)$ if $\exists u(x) \in \mathcal{E}(U)$ such that \begin{align*} \displaystyle \langle \...
2
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1answer
37 views

Bounds on Hausdorff distance via singular values

For some $\delta>0$, let $X$ and $X_\delta$ be two bounded convex polytopes in $\mathbb{R}^n$, defined as $X = \{x \in \mathbb{R}^n : Ax \leq b \}$ and $X_\delta = \{x \in \mathbb{R}^n : Ax \leq b +...
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0answers
29 views

Integration by parts, the Möbius function and the Apéry's contant

I believe that by integration by parts for $\int_0^1u\cdot dv$, see the dot here to define the formula (after we sum for a sufficently large integer) $$\sum_{k=3}^\infty\int_0^1\frac{d}{dx}x^{\mu(k)}\...
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0answers
25 views

Is this proof about a strange isomorphism correct?

Let $X$ and $Y$ be arbitrary sets and $f:X\rightarrow Y$ an isomorphism. Prove that there exist a transformation $g:Y\rightarrow X$ such that $f\circ g$ is the identity in $Y$. As X and Y havn't a ...
0
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1answer
24 views

Sum of LSC functions also LSC function

Let $f_1$ and $f_2$ are non-negative lower semicontinuous (LSC) functions on $X$. Then $f_1+f_2$ also LSC function. Proof: Let $\alpha$ be any real number then consider the set: $$E_{\alpha}=\{x: f_1(...
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0answers
20 views

Confusion on Gaussian curvature computation

Exercise I'm attempting to find the Gaussian curvature of the catenoid $M$ parametrized by $$ f(u,v)=(a\cosh v\cos u,a\cosh v\sin u,a v). $$ I've run through the typical computations of $E,F,G,e,f,g$ ...
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1answer
33 views

Putnam and beyond section 2.3.2 example

In section 2.3.2 of the book "Putnam and Beyond" there is an example problem attributed to D. Andrica, for which I think the provided proof has an error. In particular, it is asserted (on pg. 65) ...
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3answers
216 views

How do we prove that $\int_0^1 \ln x\left({1\over \ln{x}}+{1\over 1-x}\right)^2\,dx=\gamma-1?$

How do we prove that: $$\int_{0}^{1}\ln{x}\left({1\over \ln{x}}+{1\over 1-x}\right)^2\, dx =\color{blue}{\gamma-1}?\tag1$$ The only idea came to mind was this series $$\sum_{n=1}^{\infty}{1\...
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2answers
63 views

Proof that $1/f(x)$ is bounded

I wish to prove the following theorem. Statement: If $$\lim_{x\to \ a} f(x)=b\neq0,$$ then the function $$\frac{1}{f(x)}$$ is bounded for $x\rightarrow a.$ Attempt From the definition of a ...
4
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3answers
107 views

Find the closed form for $\int_{0}^{1}\left(x^k+{\ln{x}\over 1-x}\right)^2dx=f(k)$

Find the closed form for $$\int_{0}^{1}\left(x^k+{\ln{x}\over 1-x}\right)^2dx=f(k)\tag0$$ Setting $k=0$ then $f(0)=1$. $f(k)$ seems to be rational numbers for all values of $k\ge0.$ We are ...
2
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3answers
67 views

The limit of $f(x,y)= \dfrac {x^2 y}{x^2 + y^2}$ as $ (x,y) \to (0,0)$

In order to prove that the limit as $(x,y)$ approaches to $(0,0)$ of $f(x,y)= \dfrac {x^2 y}{x^2 + y^2}$ is equal to $0$ is wanted to proof: for ever $\beta\gt0$ exists some $\delta\gt0 $such that ...
2
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0answers
33 views

$f(x)$ continuous and periodic, then $f$ is bounded and reach its extreme values

I know that if $f$ has period $k$, then $f([0,k])$ is compact, since $[0,k]$ is compact. By the weristrass theorem, the function at this interval reaches its extreme values in the interval $[0,k]$. ...
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1answer
28 views

Prove/Disprove if $B$ is bounded from below so $A$ is bounded from below

Prove/Disprove: If $A\subseteq B \subseteq \mathbb{R}$ and if $B$ is bounded from below so $A$ is bounded from below Proof: Let's take $x\in A$, because $A\subseteq B$ so, for all $x\in A$, $x\in B$...
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4answers
35 views

Prove that the empty set is a subset of every set.

The exercise is taken from Rudin, principles of mathematical analysis, chapter 2 ex. 1. Let $A$ a set and let also $B$ such that $A \cap B = \emptyset$ This implies: $$ \emptyset = A \cap B \...
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0answers
47 views

Prove: There is no $r\in \mathbb{Q}: r^{n}=p$ where $n\in \mathbb{N}$, p is prime

Let assume that there is such $r\in \mathbb{Q}$, and that $r$ is irreducible fraction $\frac{m}{z}$. $r^n=(\frac{m}{z})^n=p\iff \frac{m^n}{z^n}=p \iff m^n=p*z^n$ Because $m^n$ can be divided by $p$ ...
0
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0answers
19 views

About continuity of scalar fields.

Using the usual definition of limits, with "epsilon and deltas", how can I show that if $x=(x_1,\dots,x_n)$ is a vector in $R^n$, and $f\colon J\to R$,where $R$ is the set of real numbers and $J$ is a ...
0
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1answer
31 views

Checking proof of simple number theory problem

I came up with a solution to a number theory problem. Please check it. Prove that $a^2 + b^2 + c^2 + d^2$ is never a prime if $ad=bc$, where $a,b,c,d$ are positive integers. We will prove the more ...
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1answer
69 views

As a reviewer of a math manuscript do you accept graph of a function as a proof for an inequality?

Let's say I have a function $f(x)$ and an empirical approximation of the function $\tilde{f}(x)$. I cannot prove mathematically that errors is in certain bound. However, when I plot the error, the ...
0
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0answers
28 views

If $D \subset \mathbb R^n$ convex, then $\int_{\left \| y \right \|=1} \frac{dy}{\text{distance along y from x to } \partial D}$ is constant in $x$

Let $D$ be a convex region, and let $x \in D$. for every unit vector $y$, let $d_y(x, \partial D)$ denote the distance from x to $\partial D$, measured along the ray connecting $x$ to $y$. Then it is ...
1
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2answers
34 views

tangent to a level surface

Let $F:\mathbb{R}\to \mathbb{R}^n$ be differentiable. Let $f:\mathbb{R}^n \to \mathbb{R}$ be continuously differentiable and such that the composition $g(t)=f(F(t))$ exists. If $F'(t_0)$ is tangent ...
1
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1answer
23 views

Proving i-th Fibonacci number by induction, can an inductive step be used for two sequential values?

I am working through the beginning of Introduction to Algorithms, and came across the problem Prove by induction that the $i$-th Fibonacci number satisfies the equality $$ F_{i} = \frac{\phi^{...
1
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1answer
41 views

Computing the homology of the torus with coefficients in $\Bbb F_p$, using two methods

I have some trouble to compute the homology of the torus with coefficients in $\Bbb F_p$ for $p$ a prime number. In particular I have a problem for $H_1$ : 1) The first way to compute it is to use ...