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

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3answers
60 views

Prove $\forall n \in \mathbb{N}$ where $ n \neq 1 , n + \frac{1}{n} > 2$ using completing the square.

I have got this far; I am only unable to understand how to finish the proof. $n>0 \implies n + 1/n > 0 \implies n + 1/n + 2 - 2 > 0 \implies {\big(\sqrt{n}+\frac{1}{\sqrt{n}}\big)}^2 - 2 >...
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3answers
39 views

Show finite complement topology is, in fact, a topology

My attempt to prove the following is below: Let X be an infinite set. Show that $\mathscr{T}_1=\{U \subseteq X : U = \emptyset $ or $ X\setminus U $ is finite $ \}$ My book calls this set the "...
2
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4answers
77 views

Proof that $A \cap B$ and $A \setminus B$ are disjoint.

I am trying to prove that $A \cap B$ and $A \setminus B$ are disjoint. Here is what I've done so far. Is there anything that's wrong in my proof, and is there anything that can make it better? ...
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1answer
59 views

Nonempty closed sets on a connected space imply nonemptiness of intersection?

I am dealing with just real line to make things little easier for me. Suppose we have a set $X=[0,x],X'=[x,\infty)$. For the sake of argument, assume both are closed and nonempty. Claim: By the ...
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1answer
17 views

Function exercise check-up

I want to make sure I did everything correctly, so here's the exercise: Given $P$ the set of positive prime numbers and be $S = \mathbb N^* - \{1\}$. $\forall n \in S,\ \pi(n)$ is the set of the ...
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1answer
44 views

Analytic function $f$ satisfying $f=f \circ g$

In this question, Is there an analytic function with $f(z)=f(e^{iz})$?, it was settled that there exists no non-constant analytic function $f$ such that $f=f \circ g$, where $g(z)=e^{iz}$. Below is an ...
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1answer
120 views

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable?

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable? To clarify: the problem stated that the composition is well ...
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1answer
42 views

Properties of the power set of $A$

Let $A$ be any set . Let $\wp(A)$ be the power set of $A$. Then which of the following are true 1) $\wp(A) = \emptyset$ for some $A$ 2) $\wp(A) $ is a finite set for some $A$ 3) $\wp(A)$ is a ...
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0answers
30 views

Proving $J_n(x)N_{n+1}(x)-J_{n+1}(x)N_n(x)=-\dfrac{2}{\pi x}$: Part $3$ of $3$

This is the final part of a calculation that proceeds from this previous question. Here is almost a word for word copy of the textbook question: Use the recursion relations below (for the $N_n(x)$...
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0answers
24 views

Proof explanation: diam(cl(E)) = diam ( E)

Yesterday I posted a question on this site about proving diam($E$) = diam(cl($E$)). A user posted his proof but there was some part that I don't get, I commented on his proof but he didn't respond ...
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1answer
33 views

Doubt in the correctness of the proof by induction of the corollary to the Fundamental Theorem of Algebra

I came across the following proof of the corollary of the Fundamental Algebra Theorem, which I shorten as follows: "Every polynomial $p(z) = a_nz^n + a_{n-1}z^{n-1}+...+a_1z+a_0$ has a factorisation ...
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3answers
54 views

Method of Proof in Showing Something is Smallest (Subspace)

I am reading a proof that shows the sum of subspaces is the smallest subpsace containing all the summands (It is a vector space over $\mathbb{R^n}$). The author of the book goes to show first it is a ...
2
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3answers
58 views

Prove $f(b)\equiv f(c)$(mod $m$)

Is this sufficient in proving the following statement? Also is there a more efficient way of doing so? Thanks in advance. Prove: If $f(x) = a_nx^n +\dots +a_1x+a_0$ is a polynomial with integer ...
1
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1answer
28 views

Find the minimum, maximum, minimals and maximals of this relation

Tell if the following order relation is total and find the minimum, maximum, minimals and maximals: $$\forall a,b \in\mathbb Z,\ \ a\ \rho\ b \iff a \leq b\ \text{ and }\ \pi(a) \subseteq \pi(b)$$ ...
2
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1answer
97 views

How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? [duplicate]

I am learning proofs and a question was posed which asked us to prove that $\sqrt{2}^{\sqrt{2}}$ is irrational. They mentioned this - Hint: try using the log10 function... I tried my hand at the ...
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1answer
44 views

Singularities of $z\mapsto\frac{z}{\mathrm e^z-1}$ and the Bernoulli numbers of its expansion

Characterize all singularities of $$z\mapsto\frac{z}{\mathrm e^z-1}.$$ What is the radius of convergence of the taylor expansion of $$\frac{z}{\mathrm e^z-1}=\sum_{n=0}^\infty B_n\frac{z^n}{n!}$$ ...
1
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1answer
42 views

How many employees are there in the office

In an office, $37$ people like $X$ drink, $32$ people like $Y$ drink and $41$ people like $Z$ drink. Also, $5$ employees like all the types of drinks and $15$ people like at least two types of drinks. ...
4
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2answers
88 views

problem proving: $(1+q)(1+q^2)(1+q^4)…(1+q^{{2}^{n}}) = \frac{1-q^{{2}^{n+1}}}{1-q}$

I'm trying to prove this, and it is really frustrating, because it seems a really easy problem to prove, however, I'm having a little problem with exponents: $$(1+q)(1+q^2)(1+q^4)...(1+q^{{2}^{n}}) = ...
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2answers
57 views

Disprove a limit using epsilon definition

Disprove: If $\{x_n\}$ is any sequence, and $\{s_n\}$ is a sequence converging to $0$, then $$\underset{n \rightarrow \infty}{\lim} x_n s_n=0$$ I'm not sure how to go about solving this proof. I ...
4
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0answers
61 views

Prove that $f$ is invertible

Did I show enough to prove $f$ is invertible? Alternatively is there a more efficient way to do so? Thanks in advance for any help. Let $f : X \rightarrow Y $a nd $g : Y \rightarrow X$ be ...
5
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2answers
72 views

Let $f_n$ integrable in $[a,b]$ in all $n$. Show that if $(f_n)\to f$ uniformly in $[a,b]$ then $f$ is integrable in $[a,b]$

I want a check of this proof because I'm not completely sure about the manipulation in some inequalities. Let $f_n$ integrable in $[a,b]$ in all $n$. Show that if $(f_n)\to f$ uniformly in $[a,b]$ ...
1
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1answer
36 views

Adding coins and Multiplication Game

I am currently creating a maths game and need a little help. Inside the Game you can add up coins. maximum 1 each (1p,2p,5p,10p,20p,50p,£1,£2) You can also multiply the current sum maximum 1 each ...
1
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2answers
50 views

Proof verification: diam(E) = diam(closure(E))

Since $E \subseteq cl(E) $, then it is immediate that diam $(E) \leq $ diam(cl($E))$. I only need to show that assuming diam $(E) < $ diam(cl($E))$ will lead to contradiction then I can conclude ...
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2answers
44 views

Proof that the limit exists using polar coordinates

I have come to conclusion that the most efficient and thorough way to prove whether or not a limit exists in three dimensions is to use polar coordinates. $lim_{x,y \to (0,0)} \frac{x^3+y^3}{x^2+y^2}...
2
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6answers
100 views

Three positive numbers a, b, c satisfy $a^2 + b^2 = c^2$; is it necessarily true that there exists a right triangle with side lengths a,b and c?

If so, how could you go about constructing it? If not, why not? I am new to proofs and I was reading a book where they posed this question. I understand that if we are given any right triangle, the ...
1
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1answer
20 views

Questions about the proof: Continuous function with weak derivatives $\Rightarrow$ $C^1$

For an open set $\Omega$ of class $C^1$, suppose we have $u \in W^{1,p}(\Omega)$ and that $u$ is continuous and all the partial derivatives of $u$ are continuous. I want to show that $u$ is $C^1(\...
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1answer
37 views

Finding the length of a triangle using Sine Law

I am having trouble solving, for this triangle. I am trying to find RS using the Sine Law So, $${a \over Sin A}= {b \over Sin B} \\ {25.6 \over Sin 120} = {b\over Sin 28} \\ {25.6 \over Sin 120} \...
2
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1answer
40 views

On the matter ; If $f:X \to Y$ is a function with closed graph and compactness preserving then $f$ is continuous

Let $X,Y$ be metric spaces , $f:X \to Y$ be a function , with closed graph , carrying compact sets to compact sets ; then I claim that $f$ is continuous Proof: Let , if possible , $f$ be not ...
1
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1answer
24 views

Speciel orthogonal group

I'm trying to solve 3 problems on special orthogonal groups, and I need proof verification of the first 2 and help with the proof of the 3rd. Consider $SO(n)$ the set of all $n \times n$ matrices ...
0
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0answers
50 views

Splitting field of $x^3 - 3x + 1 \in \mathbb Q[x]$

Determine the splitting field of the polynomial $p(x) = x^3 - 3x + 1 \in \mathbb Q[x]$. I verified that $p(x)$ is irreducible over $\mathbb Q[x]$. Then $(p(x))$ is maximal and $\mathbb Q[x]/(p(x))$ ...
0
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1answer
30 views

Prove for some $n$, $f^{n+1}=f^n$ and that Y is bijective.

Are these sufficient to show what is being asked? If you could confirm or provide a more efficient way to do so I would greatly appreciate it. Let $X$ be a finite set and $f:X\rightarrow X$ be a ...
1
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4answers
57 views

Help with proving if $s_n$ converges to $0$ and $x_n$ is a bounded sequence, then $\lim(x_ns_n)=0$ [duplicate]

Prove: If $s_n$ converges to $0$ and $x_n$ is a bounded sequence, then $\underset{n \to \infty}{\lim}(x_ns_n)=0$ I'm have trouble getting started on this proof. Since I know $s_n$ converges to $0$...
1
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2answers
45 views

Prove $\forall p \in \mathbb R \; :\; p \gt 0, \; \lim \frac{(-1)^n}{n^p}=0$

Prove $\forall p \in \mathbb R \; :\; p \gt 0, \; \lim \frac{(-1)^n}{n^p}=0$ So I'm very new to analysis and proofs in general, so I'm sure I did this incorrectly but here is my attempt: Suppose $\...
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2answers
40 views

Let $a,b,m,n\in\mathbb{N^*}$ with $\gcd(a,b)=1$. Is my proof correct that $(a^m,b^n)=1$?

$\gcd(a,b)=1 \iff \exists k,l \in \mathbb{N^*}(ka+lb=1) $, by Bezout's identity. Suppose $k=a^{m-1}\in \mathbb{N}$ and $l=b^{n-1}\in \mathbb{N}$. Then $ka+lb=a^{m-1}a+b^{n-1}b=a^m+b^n=1$, as ...
2
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1answer
32 views

Prove that $R\cap S$ is symmetric, transitive, and anti-symmetric.

If you can confirm these are done correctly or offer another way to do so I would greatly appreciate it. Also how would you go about proving $R\cap S$ is reflexive? What assumption if any would be ...
0
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1answer
23 views

Proof that linear difference operator, $(σ-1)^{k+1} (p) = 0$ for all $p$ $\epsilon$ $\mathbb{Q}[t]$, with $deg(p) \leq k$.

I am trying to prove that linear difference operator, $(σ-1)^{k+1} (p) = 0$ for all $p$ $\epsilon$ $\mathbb{Q}[t]$, with $deg(p) \leq k$. In this case $\sigma(t)=t+1$ and $\sigma($anything else$)=$...
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0answers
32 views

Does this family of sequences have the limit $\left(\frac{x^{2p}-y^{2p}}{2p(\ln x-\ln y)} \right)^{1/2p}$ for $p \in \mathbb{R}$?

Define the following family of one parameter sequences: $$a_0=x,~~~b_0=y$$ $$a_{n+1}=\sqrt{a_n \sqrt[p]{\frac{a_n^p+b_n^p}{2}}},~~~b_{n+1}=\sqrt{b_n \sqrt[p]{\frac{a_n^p+b_n^p}{2}}}$$ I conjecture ...
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0answers
19 views

Proving that $-\Delta+V$ on some domain is self-adjoint

This question may look as a "proof-reading" question, but what I ask is if I correctly understand the way these concepts work, by showing how I think about them. Suppose I have the following three ...
1
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2answers
113 views

Proof of inequality between sums

Let $\{y_i\}_{i=1}^N, \{z_i\}_{i=1}^N$ be two sets of real numbers s.t. $y_i, z_i \ge 0$, $\sum_{i=1}^N y_i = 1$, $\sum_{i=1}^N z_i \le 1$. I have been asked to show that $$ \sum_{i=1}^N y_i \log \...
1
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1answer
66 views

If $\sum a_n$ converges absolutely, then $\sum (e^{a_n} - 1)$ converges absolutely.

Let $a_n$ be a sequence of real numbers (note necessarily positive) such that $\sum a_n$ converges absolutely. Prove that $\sum e^{a_n}-1$ converges absolutely My attempt: We use Cauchy's criterion. ...
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0answers
26 views

How to prove that three different modulo 9 equations results the same sequence?

First let index sequence $ℕ_0=(0,1,2,…)$ and $n∈\mathbb{N}_{0}$. Then let: $$S_a = (-1)^n(a+bn) \text{ mod 9 } \text{ where } a = 1\text{, } b = -3$$ $$S_b = 2^n \text{ mod 9 }$$ $$S_c = F_{a+bn} \...
2
votes
1answer
35 views

Verification of proof, interior is open.

Let $(\mathbb{X},d_{\mathbb{X}})$ be a metric space and $\emptyset \neq A \subseteq \mathbb{X}$ its subset. Prove that the interior $A^{\circ} =\{ a \in A | \exists\epsilon(a) > 0, B_{\epsilon}^{d_{...
1
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1answer
44 views

If $a, b, c$ are integers then $a + b + c$ is even if and only if $0$ or two of $a, b, c$ are odd.

I attempted to prove this by contrapositive but was not sure if that is the most efficient way to do so. Any feedback would be greatly appreciated. Prove: If a, b, c are integers then a + b + c is ...
1
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0answers
36 views

Help with proof that $E(G|a < G < b) \lt E(H|a < H < b)$ for truncated normal distributions

Consider two independent normally distributed random variables with equal standard deviations, $G\sim N (\mu_{G}, \sigma)$ and $H\sim N (\mu_{H}, \sigma)$ that are truncated between points $a$ and $b$....
4
votes
1answer
26 views

Proof verification needed on exercise: a product of finitely many continuous functions $f_i:X\to\mathbb{R}$, $X$ compact and Hausdorff

I'm preparing for a qualifying exam in the fall, and I'm attempting the following exercise: Let $n\geq 1$ and $$ \big\{f_i\,:\,X\to\mathbb{R}\,|\,i=1,\ldots ,n\big\} $$ be a finite family of ...
4
votes
2answers
70 views

Proving that $f_2+f_4+\cdots+f_{2n}=f_{2n+1}-1$ for Fibonacci numbers by induction

Given: $f_1 = f_2 = 1$ and for $n \in\mathbb{N}$, $f_{n+2} =f_{n+1} + f_n$. Prove that $f_2 + f_4 + \dots + f_{2n} = f_{2n+1}- 1$. Would you start with setting $f_2 + f_4 + \dots + f_{2n}= ...
1
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4answers
57 views

Prove $\lim\frac{4\sin(n^2)}{3n}=0$

Prove $\lim\frac{4\sin(n^2)}{3n}=0$ Using the fact that $\left|s_n-s\right|\lt \epsilon$ I'm finding it difficult to solve for $n$. I recognize the function is bounded between $-1$ and $1$, but I ...
11
votes
7answers
390 views

Prove that $\int_{0}^{\infty}{1\over x^4+x^2+1}dx=\int_{0}^{\infty}{1\over x^8+x^4+1}dx$

Let $$I=\int_{0}^{\infty}{1\over x^4+x^2+1}dx\tag1$$ $$J=\int_{0}^{\infty}{1\over x^8+x^4+1}dx\tag2$$ Prove that $I=J={\pi \over 2\sqrt3}$ Sub: $x=\tan{u}\rightarrow dx=\sec^2{u}du$ $x=\...
1
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0answers
54 views

Natural isomorphism $\Gamma_\ast(\mathscr{F})^{\tilde{}} \to \mathscr{F}$

This question concerns Proposition 5.15, II, Hartshorne, which states that the natural map $\beta \colon \Gamma_\ast (\mathscr{F})^{\tilde{}} \to \mathscr{F}$ is an isomorphism of $\mathcal{O}_X$ - ...
1
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1answer
37 views

Vector subbundle and frame field relation

Question: Let $E \to M $ be a vector bundle of rank $k$. Suppose that for each $p \in M $ we are given a subspace $E'_p$ of $E_p$ and consider the set $\displaystyle E' = \bigcup_{p \in M} E'_p $....