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

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1answer
13 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
0
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3answers
102 views

$1+1=2$…but Why? [duplicate]

A study that was carried on recently showed that even babies at the age of few months know that $1+1=2$. My question is : is this a fact that can be proved, or is it a just a postulate as those in ...
0
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1answer
25 views

Would the following series of implications be logically correct?

Let $a$ and $b$ be positive integers, and let $f$ be a generic function satisfying $f(1) = 1$, and taking on only positive integer values. Suppose that I have the following propositions: $$\bf{A} : ...
0
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1answer
13 views

Prove how to maximize Standard Deviation given a certain mean $\bar{x}$ and set of values

I'm talking specifically of population SD, where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ I have a hunch that $s$ is maximized for a certain mean $\bar{x}$ when the values in ...
2
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1answer
38 views

Prove that Standard Deviation is always $\geq$ Mean Absolute Deviation

Where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ and $$ M = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$$ I came up with a sketchy proof for the case of $2$ values, but I would like ...
0
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1answer
26 views

Real Analysis, Folland Corollary 2.19 Integration of Nonnegative functions

Corollary 2.19 - If $\{f_n\}\subset L^+$, $f\in L^+$, and $f_n\rightarrow f$ a.e., then $\int f \leq \liminf\int f_n$. Proof - We have that $\{f_n\}\subset L^+$, $f\in L^+$ and $f_n\rightarrow f$ a....
4
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2answers
50 views

Real Analysis, 2.18 (Fatou's Lemma) Integration of Nonnegative functions

2.18 Fatou's Lemma - If $\{f_n\}$ is any sequence in $L^+$, then $$\int \left(\lim_{n\rightarrow \infty}\inf f_n\right) \leq \lim_{n\rightarrow \infty}\inf\int f_n$$ Attempted proof - We know that $$...
1
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1answer
22 views

Show excluded point topology is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set, and $p$ be an arbitrary point in $X$. Show that $\mathscr{T}_4=\{U \...
1
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2answers
19 views

Show particular point topology, is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set, and $p$ be an arbitrary point in $X$. Show that $\mathscr{T}_3=\{U \...
2
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0answers
31 views

Show “countable complement topology” is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set. Show that $\mathscr{T}_2=\{U \subseteq X : U = \emptyset $ or $ X\...
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4answers
46 views

Prove $3\mathbb{Z}+1=\{6\mathbb{Z}+1\}\cup\{6\mathbb{Z}+4\}$

I was wondering if someone could confirm I have proven the following equality correctly. Also, for part II should I have let $n\in \mathbb{Z}$ as opposed to $n\in 6\mathbb{Z}+1$ or was I correct? ...
1
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2answers
41 views

Could someone please check my proof that $(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Show that if $f:X\to Y$ is uniformly continuous, then $$(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$$ My ...
3
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4answers
137 views

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{n}{\left(n+1\right)!} = 1-\frac{1}{\left(n+1\right)!}$ So I proved the base case where $n=1$ and got $\frac{1}{2}...
0
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0answers
11 views

Auslander-Reiten theory: exercise $23.b$ of 'Elements of the Representation Theory of Associative Algebras'

I am solving exercise $23.b$ of chapter IV of 'Elements of the representation theory of associative algebras' by Assem, Simson and Skowronski. The question is the following: Consider the following ...
3
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1answer
14 views

Prove that $\text{Dom } (S\circ R) ⊆ \text{Dom }R $

Let $R$ be a relation from $A$ to $B$ and $S$ be a relation from $B$ to $C$. Suppose, $x \in \text{Dom }(S\circ R)$. Then, it follows that there $\exists y \in C$ such that $(x,y) \in S\circ R $. ...
0
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3answers
39 views

Is this a valid existence proof for: “there exists a unique real number solution to the equation $x^3 + x^2 - 1 = 0$ between $x = 2/3$ and $x = 1$”

I was wondering if this was a valid existence proof for the following: "there exists a unique real number solution to the equation $x^3 + x^2 - 1 = 0$ between $x = 2/3$ and $x = 1$" Proof: Assume ...
0
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0answers
30 views

How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
0
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1answer
44 views

Is it my error or the term “normal” has multiple meanings?

I use a definition of normal quasi-uniform spaces from this article. Now I have proved (I do not present the proof because it uses "funcoids" which can be read about only in my manuscripts.) that ...
0
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3answers
24 views

Proof by contradiction for: Prove that there do not exist positive integers $m$ and $n$ such that $m^2 - n^2 = 1$

I am kind of stuck on a practice problem relating to proof by contradiction that goes as follows: "Prove that there do not exist positive integers $m$ and $n$ such that $m^2 - n^2 = 1$" For the ...
1
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1answer
38 views

Real Analysis, Folland Theorem 2.14 (Monotone Convergence Theorem)

Theorem 2.14 (MCT) - If $\{f_n\}$ is a sequence in $L^+$ such that $f_{n}\leq f_{n+1}$ for all $n$, and $f = \lim_{n\rightarrow \infty}f_n (=\sup_n f_n)$, then $\int f = \lim_{n\rightarrow \infty}\int ...
7
votes
1answer
59 views

A second opinion on a proof in topology

My friend and I were looking over some homework questions for an upcoming test in introductory topology, and one of the questions on the homework was to show that a metric space is normal. What we ...
1
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0answers
17 views

Sum over square divisors is multiplicative proof verification

I would like someone to verify my proof of the following claim, which I have been using to solve some problems about proving series identities in Ch. 11 of Apostol's analytic number theory text. Let $...
2
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0answers
23 views

Proof check: commutation of Galois automorphisms and complex conjugation in CM-fields

Let $K/\mathbb{Q}$ be a Galois CM-field with $Gal(K/\mathbb{Q})=:G$ and $J_\mathbb{C}$ be the complex conjugation. Since $K$ is a CM-field one can show, that $$J:=\phi^{-1}\circ J_\mathbb{C}\circ \phi=...
0
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3answers
46 views

What is the probability of getting NO PAIRS in a $13$-card poker game?

What is the probability of getting NO PAIRS in a $13$-card poker game? Here is my attempt: The setup for the required poker hand would be: $$ABCDEFGHIJKLM$$ where $A, B, \ldots, M$ are distinct ...
0
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0answers
39 views

Work required to align pieces in a plane.

Given two piecewise continuous functions f(x) and g(x) and that $\lim_{a -> x^-} g(a) - f(a) = \lim_{a -> x^+} g(a) - f(a)$ at all points, find the work used to shift each of the planar slolids ...
0
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0answers
21 views

Convergence of a big sum indexed over $\mathbb{Z}^3$

For a fixed vector $r_j$ consider the function on $\mathbb{R}^3$ defined by the series $$f(r) = \sum_{\substack{n,m,k \in \mathbb{Z} \\ (n,m,k) \neq 0}} \frac{1}{n^2+m^2+k^2}e^{2\pi i(n,m,k) \cdot (r-...
2
votes
2answers
77 views

Rotation matrix check

Let matrix $A=\frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -1 \\ 1 & 1 \\ \end{bmatrix} $. Check if $A$ is a rotation matrix in $\mathbb{R^2}$ by angle $\theta=\...
1
vote
1answer
39 views

Real Analysis, Folland Problem 2.1.5 Measurable Functions

Problem 2.1.5 - If $X = A\cup B$ where $A,B\in M$, a function $f$ on $X$ is measurable if and only if $f$ is measurable on $A$ and on $B$. Proof - Suppose, $X = A\cup B$ where $A,B\in M$ and we have ...
0
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1answer
41 views

Fibonacci Sequence: Prove $f_1+f_3+\dots+f_{2n-1}=f_{2n}$ by Induction.

I believe the majority of my proof is correct I'm just not certain about the base case if any one can explain how to do that base case or fix any error I made I would greatly appreciate it. Recall ...
0
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0answers
10 views

I need input and help understanding how the formula for x arises in a cycloid that is parameterized with theta with the cusp at the origin

Disclaimer: I attempted to answer some of it by using my own deductions. I would feedback on that. The book gives the formulas for how x arises but my problem is understanding how the formulas arose. ...
0
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1answer
29 views

Every completely regular space is regular

The definitions I'm working with: $(X,\mathcal{T})$ is said to be completely regular if for every $x \in X$ and every closed set $C \subseteq X$ not containing x can be separated by a continuous ...
13
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5answers
172 views

I want to show that $\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$

I want to show that $$\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$$ Expand $(x^4-x+\pi)^2=x^4-2x^3+2x^2-2x\pi+\pi{x^2}+\pi^2$ Let see (substitution of $y=x^2$)...
3
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4answers
36 views

How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?

Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$. One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
1
vote
1answer
15 views

Show that there exists at most one extension of $f$ whose co-domain is a Hausdorff space [duplicate]

I want to show the following Suppose $A \subset X, f: A \to Y$ is continuous, $Y$ is Hausdorff. Show that there is at most one continuous extension $g: \overline A \to Y$ I feel like I am ...
0
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1answer
28 views

Contrapositive, Negation, and Converse of statements

I am having trouble with the wording of these statements particularly the negation statement. Is that the best way to put it or could you provide a better alternative? Also for the converse proof ...
2
votes
4answers
50 views

Antiderrivative of ${d^2 y \over dx^2} = 1-x^2$

At any point $(x,y)$ on a curve, ${d^2 y \over dx^2} = 1-x^2$, and an equation of the tangent line to the curve at the point $(1,1)$ is $y=2-x$. Find an equation of the curve. This is what I've done ...
0
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0answers
19 views

Showing $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ is a Hilbert space

Let $I$ be an open interval in $\mathbb{R}$. We define $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ with the scalar product of the Sobolev space $H^1(I)$, i.e. $(u,v)=(u,v)_{L^2(I)}+(u',v')_{L^...
0
votes
1answer
38 views

$\succsim$ preorder on X being continuous imply lower contour set closed

$\succsim$ is preorder (i.e. preference relation) on X that is continuous. This implies the lower contour set is closed. Would you please share your 2 cent on my parenthesis explanation (e.g. line ...
0
votes
0answers
27 views

Eliminate the parameter to find a cartesian equation for the curves

For the first part I am just unsure as to how the book has a different answer than mine. The book has the answer $y = \frac{3}{4} x - \frac{1}{4}$ but given the functions $x(t) = 3 - 4t$ and $y(t) = ...
1
vote
1answer
36 views

Arithmetic progressions form infinite basis on $\mathbb{Z}$

Let $B(a,b) = \{ax+b: a,b \in \mathbb{Z}, a \neq 0, x \in \mathbb{Z}\}$ be a so called arithmetic progression I am required to show that that $\mathcal{B} = \{B(a,b) | a,b \in \mathbb{Z}\}$ is a ...
3
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4answers
93 views

$a^n$ even implies $a$ even

I've tried to prove that $(\forall a,n>0 \in \mathbb{N}),(a^n \text{ even} \implies a \text{ even})$, can someone tell me whether my proof is sound? Lemma 1: $a \text{ even} \implies a^2 \text{ ...
2
votes
2answers
34 views

Construction of field extension for $[E:\mathbb F_{11}]=3$

Let $\mathbb F_{11}\subset E$. Construct a field extension $E$ of $\Bbb{F}_{11}$ such that $[E:\mathbb F_{11}]=3$ Answer: Let $f(x)=x^3+1 $ be a polynomial in $\mathbb F_{11}[x]$ with $deg(f)=3$. ...
16
votes
3answers
333 views

Prove $\pi^2\int_0^\infty\frac{x\sin^4\pi x}{\cos\pi x+\cosh\pi x}dx=e^2\int_0^\infty\frac{x\sin^4ex}{\cos ex+\cosh ex}dx=\frac{176}{225}$

Marco Cantarini and Jack D'Aurizio proved hard-looking integrals (see Marco and Jack) in my recent two posts. This is our final hard-looking integral that yield a rational answer: $$\pi^2\int_{...
2
votes
1answer
40 views

Is this proof of convergence in probability correct?

${X_i}, i = 1,2,\dots$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $\frac{S_n}{n} \to 0 \quad $ in probability show that $$\lim_{n\to \infty} \min_{...
2
votes
1answer
31 views

Proof by contradidction that the mean of a set cannot be greater than the greatest value in that set.

I want to prove that given a set of values $x_1, x_2, ..., x_n$, the mean of those values cannot be greater than the greatest of those values. Let the mean $\frac{x_1 + x_2 +... + x_n}{n} = a$ ...
0
votes
0answers
48 views

Unusual integration of 1/cx [duplicate]

Consider an integral: $$\int_2^3 \frac{1}{cx} dx$$ where $c$ is a constant So we can take that out of the integral, so $$\int_2^3 \frac{1}{cx} dx = \frac{1}{c} \int_2^3 \frac{1}{x} dx $$ all is ...
3
votes
1answer
53 views

Let $g_n(x)=[x\in (r_n,1]]$. Show that $G(x)=\sum_{n\ge 1} g_n(x)/2^n$ is Riemann-integrable

I want to check if my proof about $G(x)$ is Riemann-Integrable in $[0,1]$ is correct. Let $\{r_n\}$ an enumeration of the rationals in $[0,1]$, and $g_n(x)=[x\in (r_n,1]]$. Show that $G(x)=\sum_{n\...
2
votes
5answers
136 views

Prove: if $|x-1|<\frac{1}{10}$ so $\frac{|x^2-1|}{|x+3|}<\frac{1}{13}$

Prove: $$|x-1|<\frac{1}{10} \rightarrow \frac{|x^2-1|}{|x+3|}<\frac{1}{13}$$ $$|x-1|<\frac{1}{10}$$ $$ -\frac{1}{10}<x-1<\frac{1}{10}$$ $$ \frac{19}{10}<x+1<\frac{21}{10}$$ $...