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Questions tagged [proof-verification]

For questions concerning a specific proof or a specific solution, asking for verification, identifying errors, suggestions for improvement, etc. (You should not use this tag if the question does not contain a proposed proof/solution.)

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Formally deriving a vacuous truth from a definition involving conjoined implications

Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Longrightarrow |x| = x) \wedge (x < 0 \Longrightarrow |x| = -x)$ I want to use this definition in one of my proofs. So I ...
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3answers
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Question on the injectivity of a function.

Consider the map $f\colon\mathbb{S}^1\to\mathbb{S}^1$ defined as $f(z)=z^2$, where $\mathbb{S}^1$ is the unit circle, $$\mathbb{S}^1=\{z\in\mathbb{C}:|z|=1\}.$$ On $\mathbb{S}^1$ we define the ...
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0answers
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Proving an inequality involving absolute value; how do I justify using a conjunction (and) instead of a disjunction (or)?

I'm putting together the following the proof, and I have a question about one of the final steps. Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Rightarrow |x| = x) \...
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0answers
24 views

$f_n\to f$ in $L^1(\mu)\implies f_n $ are uniformly integrable

If $f_n\to f$ in $L^1(\mu)\implies f_n $ are uniformly integrable where $\mu $ is positive measure My Attempt: Uniformly Integrable family: $\{f_n\}_{n\in A}$ is said to be uniformly integrable if $...
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5answers
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Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
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1answer
25 views

proof that $Y$ follows normal distribution

I'm new to probability and studying multivariate normal distribution. The one thing I don't understand is the linear transformation of multivariate normal distribution.If the $X$ follows Normal ...
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2answers
46 views

Transform $dx/dt$ to $dr/dt$ polar coordinates

I've had to screenshot the question and post it as a photo
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1answer
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Probability the event occurs knowing that I received no information

First I want to thank you if you pay attention to my post. I apologize if it seems elementary to you, note that I searched a lot an answer before posting. I'm going to try not to be vague, do not ...
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1answer
49 views

Proving that $\lim_{(x,y)\to(0,0)} (x^2+y^2)^{x^2y^2}=1$ using polar coordinates

Am I doing this right? I rewrite the function as follows: $$(r^2\cos^2\theta+r^2\sin^2\theta)^{r^4\cos^2\theta\sin^2\theta} \stackrel{\text{various trig identities}}{=} r^{\frac{1}{4}r^4\sin^2 2\...
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0answers
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Proof Verification: Classify all groups of order 6

I'm currently in an introductory group theory class, and I'm supposed to classify all groups of order 6 up to isomorphism without use of more advanced concepts like the Sylow theorems or even the ...
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Substituting matrix with diagonal form in problems

I was trying to solve the following problem (SEEMOUS 2019 Problem 3): Let $A,B$ be complex-valued $n \times n$ matrices such that $B^2 = B$. Show that $$\text{rank}(AB - BA) \leq \text{rank}(AB + BA)...
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0answers
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Proof verification. Show that $\mathcal{O}(\mathcal{O}(f)) = \mathcal{O}(f)$

Prove that: $$ \mathcal{O}(\mathcal{O}(f)) = \mathcal{O}(f) $$ I've started with letting some $u \in \mathcal{O}(\mathcal{O}(f))$, then: $$ |u| \le k_1|v| $$ Where: $$ |v| \le k_2|f| $$ Also: $$ \...
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0answers
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Does this proof of continuity at irrationals of the Thomae function make any sense?

Thomae's function $ f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = 0$ if $x$ is irrational and $f(x) = \frac{1}{q}$ if $x = \frac{p}{q}$. The function $f$ is continuous at all the irrational ...
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3answers
107 views

Proof verification: Prove $\sqrt{n}$ is irrational.

Problem Let $n$ be a positive integer and not a perfect square. Prove $\sqrt{n}$ is irrational. Proof Consider proving by contradiction. If $\sqrt{n}$ is rational, then there exist two coprime ...
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1answer
57 views

Calculate $\int_{\pi\over 2}^{{\pi\over2}+i} \cos2 z dz$

Calculate $\int_{\pi\over 2}^{{\pi\over2}+i} \cos 2z dz$. I want to verify my answer please. My solution: Because $\cos 2z$ is analytic everywhere, we just need to calculate the integral: $$ \...
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3answers
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Proof that $\| u \|=\| v\|\iff\langle u+v,u-v\rangle=0$

Let $(\cdot,\cdot):\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}^n$ be any map that fulfills the properties $u,v,w\in\mathbb{R}^n;\; \lambda\in\mathbb{R};\; \langle u;v\rangle:=(u_1v_1)+\cdots+(...
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2answers
52 views

Irritating “proof” of the Collatz Conjecture

I recently stumbled across this self-proclaimed proof of the Collatz Conjecture. It seems very irritating to me that this very hard conjecture is supposedly proven by using very basic counting ...
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1answer
25 views

Limiting behavior of a meromorphic function

Refer the following answer. Can someone explains how can this follows from Morera's theorem. Thanks in advance.
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0answers
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Expressing $\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}+S(x,t)$ as an integral involving $S(x,t)$

I am trying to express the solution of$$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}+S(x,t) \tag{1}$$ for $-\infty< x<\infty$ with initial condition $u(x,0)=0$ as an integral ...
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1answer
18 views

Limit of a sequence multiply by a constant

If the sequence $s_n$ converges to $s$, then for all $k \in \mathbb{R}$ the sequence $ks_n$ converges to $ks$. Proof: We assume that $k \neq 0$ since the result is trivial for $k = 0$ . Let $\...
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1answer
46 views

Product of compact metric spaces is again compact.

Proof Let $(X_j,d_j)$ be a compact metric space for $j=1,...,n$. Denote $X=X_1 \times X_2 \times...\times X_n$ to be the product of compact metric spaces. Assume the property that the open sets in $(...
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1answer
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Is my proof of $\binom{m+n}{r}=\binom{m}{0}\binom{n}{r} + \binom{m}{1}\binom{n}{r-1} + … + \binom{m}{r}\binom{n}{0}$ right?

As the title says, I was requested to prove $\binom{m+n}{r}=\binom{m}{0}\binom{n}{r} + \binom{m}{1}\binom{n}{r-1} + ... +\binom{m}{r}\binom{n}{0}$ I was requested to do this using the following ...
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0answers
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$A$ $9\times 9$ real matrix such that $A^3=0$ and $A^{i}\neq 0$ for $i=1$ and $2$. what is the number of possible J.C. Forms(upto similarity) of $A$.

Let $A$ be a $9\times 9$ real matrix such that $A^3=0$ and $A^{i}\neq 0$ for $i=1$ and $2$. Then I want to know what is the number of possible J.C. Forms(upto similarity) of $A$. We can see that ...
2
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1answer
30 views

A step in a proof that $M, N\unlhd G$ implies $MN\unlhd G$.

This is an easy question for me to verify, I suppose, but, for whatever reason, I'm having a crisis of confidence. This is a proof-verification question. The bit I'm unsure of is highlighted in bold ...
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1answer
40 views

Counterexample: linearly ordered sets for which there exists more than one isomorphism

In my axiomatic set theory notes, there appears that, if $A$ and $B$ are well-ordered isomorphic sets, then there exists one isomorphism between them. However, as a side note, it is stated that this ...
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0answers
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Prove that $\textbf{Int}(Y)$ is the largest open set contained in $Y$

In Green and Gamelins book "An Introduction to Topology" there is an exercise Id like some feedback on. If $Y$ is a subset of $X$, then prove that $\textbf{Int}(Y)$ is the union of all open subsets of ...
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3answers
102 views

Function $F$ is surjective if and only if $F$ is $1-1$ [duplicate]

While I was working on proofs of functions, the following claim occurred to me that I think it is correct but I could not prove it. Please note that the claim may not be correct since it is just my ...
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1answer
29 views

Equivalence with measurable functions and spaces

i have been reading Follands Real Analysis Book and i got stuck with one exercise. It says that if $\lambda(X)$ is finite and $(X,M,\lambda)$ is a measure space and $(X, \overline{M}, \overline{\...
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1answer
32 views

Prove that topological space $ X= {[0,1]}^{2} $ with dictionary order topology is not second countable. (my solution)

Prove that topological space $ X= {[0,1]}^{2} $ with dictionary order topology is not second countable. I would assume that it is second countable. So then there would exist a basis $B$ of $X$ which ...
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2answers
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Let $\{a_n\}_n$ be a sequence and suppose $\{a_n\}_n$ isn't bounded above. Prove that there's a subsequence $\{a_{n_k} \}_k$ such that $a_{n_k} → ∞.$

I just wanted to see if my proof worked or not, or if there was any way to improve it. Proof: In the case that $\{a_n\}_n$ is bounded below, we have $a_n\to\infty$ and so for any subsequence $\{a_{...
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1answer
11 views

Proof Check: Outer Regularity of Lebesgue Measure

I am trying to prove that for any bounded set $A$ in the borel $\sigma$ algebra, that for the Lebesgue measure $m$ $$ m(A)=\inf\{m(U)|U\text{ is open and }A\subseteq U\} $$ here is my attempt. Let $...
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1answer
69 views

Correct non-inductive proof of Euler's Formula $|V|+|F|-|E|=2$?

Theorem: The number of vertices $|V|$, edges $|E|$, and faces $|F|$ in an arbitrary connected planar graph are related by the formula $$|V|+|F|-|E|=2$$ Proof Attempt: (For acyclic planar graphs) Let ...
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2answers
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Trig Identity Question Finding Value of K

If $\sin(x) + \cos(x) = k$ for what value(s) of $k$ can $\sin(x)\cos(x)=1$?
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0answers
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Mathematical Notation for Solution

I'm trying to write the solution to one of my problems, but am having a difficult time writing it out mathematically. I have a matrix $C$ that is given by $$ C = [A^{N-1}B,A^{N-2}B,...,AB,B]\in\mathbb{...
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1answer
32 views

Why is this argument valid? Velleman

Why is the pictured argument valid? Velleman in this chapter section says that an argument is valid only if the conclusion has the option of not being true if all the premises are true. But row 7 is ...
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0answers
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elementary matrices row operation proof

Have to prove the multiplication EA with a elementary matrix E ∈ K^m×m results in E := Em,[zr→λzr] : multiplication of row r with λ ∈ K. With A and B ∈ K^m×n And C ∈ K^n×p Same for E := Em,[zr↔zq]
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2answers
50 views

Showing that $[G:K]=[G:H][H:K]$ for $K \leq H \leq G$ and $|G| < \infty$

Definition: For a finite group $G$ and a subgroup $H \leq G$, $$[G:H] := \frac{|G|}{|H|},$$ which is a positive integer. For a finite group $G$, assume $K$ is a subgroup of $H$ and $H$ is a ...
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0answers
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Relationship between subharmonic functions and Laplacian

Show that if $u\in C^2(\Omega)$ is subharmonic then $\Delta u\geq 0$ in $\Omega$. Here is my proposed solution. I read this solution but couldn't follow every step (I am still relatively new to PDE). ...
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1answer
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proof verification: Proving an upper bound for a uniformly convergent sequence of functions on $\mathbb{R}$

Assume $f_n: \mathbb{R}\to\mathbb{R}$ is a sequence of functions that converges uniformly to $f$. Assume that there exists $M>0$ such that for all $n\in \mathbb{N}$ and $x\in \mathbb{R}$ one has $|...
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1answer
60 views

Any convergent sequence is bounded. Don’t we need to use the absolute value in this proof?

We have the following elementary result on real sequences. Any convergent sequence is bounded. This is basically the proof given in my notes: Suppose that $a_n \to a \in \mathbb{R}$. Now choose $\...
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0answers
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Let $A_1$, $A_2$ be groups and $\phi:A_1\to A_2$ surjective with $G_1\unlhd A_1$ and $\phi(G_1)=G_2$. Is $A_1/G_1\cong A_2/G_2$? [duplicate]

Let $A_1$ and $A_2$ be a group and $\phi: A_1 \to A_2$ be a surjective group homomorphism. Also, $G_1 \unlhd A_1$ and $\phi(G_1) = G_2$. I have to prove/disprove this statement: Is $A_1/G_1 \cong A_2/...
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0answers
43 views

Show that finite subgroup of units of a field is cyclic.

Let $F$ be a field. Show that every finite subgroup of $F^\times$ is cyclic. My attempt: Let $H$ be a subgroup of $F^\times$. Suppose $p\mid|H|$ with $p$ prime. Any element in $H$ of order $p$ ...
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1answer
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$\lim_{n\to\infty}, I_n=\int_{-1} ^{+1} |x|(1+\sum_{r=1} ^{2n} \frac{x^r}{r})$

Why is my method incorrect? $$\lim_{n\to\infty} I_n=\int_{-1} ^{+1} |x|(1+\sum_{r=1} ^{2n} \frac{x^r}{r})$$ Since $1>|x|$ and $\lim_{n\to\infty}$ I used the sum for an infinite geometric ...
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0answers
11 views

Soft sheaves on $T_1$-space

Suppose we have a topological space $X$ and an open subset $U \subseteq X$ with inclusion $j \colon U \hookrightarrow X$. If we have a sheaf $\mathcal F \in \text{Sh}(X)$, then we know that in ...
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1answer
10 views

Question on Partially ordered sets and images of sets

Could someone look through my attempt at proving the following problem please? Let $(A,\preceq)$ and $(B,\preceq')$ be POSETS and $C \subseteq A$. Suppose that $h:A \rightarrow B$ satisifies $x \...
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0answers
18 views

Projective dimension of quotient modules

Let $(R, \mathfrak{m})$ be a Noetherian local ring, $M$ a finitely generated $R$-module and $\underline{x}=x_{1}, x _{2}, \dots, x_{r}\in \mathfrak{m}$ an $M$-regular sequence with $r\geq 1$. If $\...
2
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0answers
54 views

Finding $\lim_{n \to \infty}\sum_{k=1}^n\sqrt{\frac{k(k+1)}{n^2+k}}$

Problem Evaluate $$\lim_{n \to \infty}\left[\sqrt{\frac{1\cdot2}{n^2+1}}+\sqrt{\frac{2\cdot3}{n^2+2}}+\cdots+\sqrt{\frac{n(n+1)}{n^2+n}}\right],n=1,2,\cdots.$$ Solution Notice that, for $k=1,2,\...
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1answer
47 views

Is the differential at a regular point, a vector space isomorphism of tangent spaces, also a diffeomorphism of tangent spaces as manifolds?

Note: My question is not "If $f$ is a diffeomorphism, then is the differential $D_qf$ an isomorphism?" My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. I didn't study much of ...
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1answer
41 views

Is there any injective homomorphism (i.e. monomorphism) from a non-cyclic group of order $4$ to $\mathbb{Z}_8$?

The only such possible group is $V$ (up to isomorphism). If $\phi$ be such an into homomorphism, then $o(\phi(V))=4$ and $\phi(V)$ being a subgroup of $\mathbb{Z}_8$, it must be cyclic with a ...
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0answers
26 views

A closed subset $Z⊂\operatorname{Spec}R$ is irreducible if and only if $Z=Z(p)$ for a uniquely determined prime ideal $p$.

Definitions I am working with: $R$ is a commutative ring with $1$. A topological space $X$ is called irreducible if every decomposition $X=X_1∪X_2$ in closed subsets $X_i$ implies that either $X_1$ ...