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Questions tagged [solution-verification]

For posts looking for feedback or verification of a proposed solution. "Is this proof correct?" is too broad or missing context. Instead, the question must identify precisely which step in the proof is in doubt, and why so. This should not be the only tag for a question, and should not be used to circumvent site policies regarding duplicate questions.

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Proof of Linear-Algebra (matrices). [duplicate]

Prove that \begin{align} \begin{pmatrix} \lambda & 1\\ 0 & \lambda \end{pmatrix}^n &= \begin{pmatrix} \lambda^{n} & n\lambda ^{n-1}\\ 0 & \lambda^{n} \end{pmatrix} . \end{align} ...
DEMB's user avatar
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3 votes
3 answers
217 views

The method of substitution in the problem of finding the integral

My teacher gave me a simple problem: Find $\int \dfrac{x}{\sqrt{x+1}} \, dx$. This is how I approached it: I set $u = \sqrt{x+1}$, which implies $u^2 = x + 1$, thus $2u \, du = dx$. Therefore, $$ \int ...
Math_fun2006's user avatar
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0 answers
22 views

The unit disc contains finitely many disjoint dyadic square whose area is arbitrarily close to the area of the disc

This is Exercise 1.25.b in Pugh’s Real Mathematical Analysis. In this post, I showed that the unit disc contains finitely many dyadic squares whose total area is arbitrarily close to the area of the ...
Number11's user avatar
1 vote
1 answer
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Solution verification for a proof that $x^3+py^3+p^2z^3=0$ has no nontrivial solution

I was asked to prove that given a prime $p\geq 3$, the equation $x^3+py^3+p^2z^2=0$ has exactly one solution in the integers. Here is my proof: Clearly $(x,y,z)=(0,0,0)$ is a solution. Let's assume ...
MSEU's user avatar
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0 votes
0 answers
19 views

The unit disc contains finitely many dyadic squares whose total area is arbitrarily close to the area of the disc

Exercise 1.25.a in Pugh’s Real Mathematical Analysis states that Given $\epsilon > 0$, show that the unit disc contains finitely many dyadic squares whose total area exceeds $\pi - \epsilon$, and ...
Number11's user avatar
1 vote
0 answers
15 views

Passing from $\mathcal{D}$ to $\mathcal{S}$ using a density argument and extra condition

I saw a proof for the following statement for the space of Schwartz functions $\mathcal{S}$: $$ \varphi \in \mathcal{S}: \int \varphi = 0 \iff \exists \Phi\in \mathcal{S}: \Phi' = \varphi, $$ which ...
Taleofwoe's user avatar
  • 109
0 votes
0 answers
13 views

Proving statement about cumulative distribution function

I want to prove the following statement: Let F fulfill the properties of a cumulative distribution function. Define $$X^-(\omega) = \inf \{z \in \mathbb{R}: F(z) \geq \omega\} \quad X^+(\omega) =\inf ...
TeX_User's user avatar
0 votes
2 answers
45 views

Existence of certain function satisfying certain conditions

I want to show that there does not exist $f ∈ C([0, 1], R)$ satisfying the following two conditions: (i) $\int_{0}^{1} f(x) dx = 1$; (ii) $\lim_{n\to\infty} \int_{0}^{1}f(x)^n dx = 0.$ Suppose there ...
Ricci Ten's user avatar
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1 vote
0 answers
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Prove carefully $C^1[0,1]$ is incomplete

This post shows how to prove $C^1 [0, 1]$ is incomplete in the uniform norm. But I want to get a deeper understanding, specifically how to come up with an example. Here's my understanding: I know $C^0[...
HIH's user avatar
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0 votes
1 answer
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Euclid's SSA proof (book VI prop 7)

I've been studying Euclid's Elements for the past few weeks, and have came across a conflicting theorem he proved. "if two triangles have one angle equal to one angle, the sides about other ...
thatpithere's user avatar
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0 answers
35 views

polynomial $f$ having $n$ distinct roots implies either $f(x)\neq 0$ or $\deg f\geq n$

This is about two different results I have read in the note, where it says Theorem 5. If $f(x)\in F[x]$ has distinct roots $a_1, a_2,\dots, a_n$, then $f(x)$ is divisible by $(x − a_1)(x − a_2) · · · ...
Mr.MathDoctor's user avatar
1 vote
1 answer
35 views

Proving the Nested Compact Set Property

There’s this theorem in my analysis book that I want to prove, which states that: If $k_{1} \subseteq k_{2} \subseteq k_{3} \subseteq ...$ Then the intersection $\bigcap_{n=1}^{\infty}k_{n} \neq \...
SpectralTheoryFan's user avatar
3 votes
4 answers
140 views

Understanding the implication in linear algebra regarding vectors

Let $V$ be a subspace of $\mathbb{R}^n$ with the usual dot product, and let $\mathbf{z}, \mathbf{w} \in V$ be fixed vectors. If for every $\mathbf{v} \in V$ it holds that $\mathbf{z} \cdot \mathbf{v} =...
brodar's user avatar
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1 vote
1 answer
80 views

$f:A\subseteq \mathbb{R}\to \mathbb{R}^2$ maps closed sets to closed sets.

Can you take a look at my proof, please? Let $f:A\subseteq\mathbb{R}\to \mathbb{R}^2$ given by $f(x)=(x,x^2)$ where $A$ is a closed set and $f$ is continuous in $A$. Show that $f$ maps closed sets to ...
Roma_Rayado's user avatar
-2 votes
0 answers
46 views

Proof the number of nodes in a full binary tree +1 is equal to the double of the leafs [closed]

This is a class problem from the book https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-spring-2015/resources/mit6_042js15_textbook/ Page 191, I believe I have done it correctly but ...
Jery Lazman's user avatar
1 vote
0 answers
22 views

Inequation regarding the relationship between |V(G)∣ and the girth of a 3-connected graph

(Dis)prove the following inequation: |V(G)∣ ≥ 2 g(G) - 2 G is a 3-connected Graph g(G) is the length of the shortest cycle in G I had some approaches to the problem but the most promising was: Let ...
Barren_Wuffet's user avatar
2 votes
1 answer
57 views

Finding the different zeros of a continuous function

I'm working on Spivak's Calculus and am doing Problem 4 of Chapter 8. Here's the problem: Suppose $f$ is continuous on $[a,b]$ and that $f(a) = f(b) = 0$. Suppose also that $f(x_0) > 0$ for some $...
Aryaan's user avatar
  • 273
0 votes
0 answers
28 views

What is wrong in this proof and where is it needed that $R$ is an integral doamin? [duplicate]

I wanted to prove the following theorem myself: Let $R$ be an integral domain and $p \in R$. If $(p)$ is a maximal ideal, Then $p$ is a prime element. My attempt: Since $(p)$ is maximal then there ...
Physor's user avatar
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2 votes
0 answers
65 views

A very interesting function equation $f(a,b)=f(a,c)+f(c,b)$ implies $f(a,b)=g(a)-g(b)$.

Function equation: $f(a,b)=f(a,c)+f(c,b)$ for all positive reals $a>c>b\geq 0$. My solution: $f(a,c)=f(a,b)-f(c,b)$, Let $b=0$, $f(a,c)=f(a,0)-f(c,0)$. Define $g(a)=f(a,0)$. We get the answer. ...
dodo's user avatar
  • 818
0 votes
2 answers
43 views

A convergence property for iid sequence of Cauchy random variables

A real random variable ${X}$ is said to have a standard Cauchy distribution if it has the probability density function $\displaystyle {x \mapsto \frac{1}{\pi} \frac{1}{1+x^2}}$. If ${X_1,X_2,\dots}$ ...
shark's user avatar
  • 929
1 vote
1 answer
67 views

Calculate the $p$-adic absolute value of $\frac{1}{p^7-p^5+p^3}$

I want to solve the following exercise: Calculate the $p$-adic absolute value of $\frac{1}{p^7-p^5+p^3}$. My Approach: For the $p$-adic absolute value, we have the following rules: $|xy|_p=|x|_p |y|_p$...
NTc5's user avatar
  • 553
1 vote
1 answer
43 views

The product of a Dedekind cut and its inverse equals one

Exercise 1.11 of the textbook Real Mathematical Analysis by Pugh asks what the inverse of a cut $x=A|B$ is ($x$ being positive). The first step is to show that $x^*=C|D$ where $C=\{1/b: b\in B \space\&...
Number11's user avatar
1 vote
0 answers
41 views

Proof of factor theorem regarding polynomials

Division of polynomials: Let $f$ and $g$ be polynomials with $g(x)\neq 0$. Then there exist unique polynomials $q$ and $r$ such that $r=0$ or $\deg r<\deg g$, and $f(x)=q(x)g(x)+r(x)$. I want to ...
Mr.MathDoctor's user avatar
3 votes
2 answers
62 views

$A \cup C \subseteq B \cup C$ iff $A\setminus C\subseteq B\setminus C$

I have attempted to prove the theorem, but I'm not sure whether I can invoke proof by contradiction on the converse the way I did or not. $A \cup C \subseteq B \cup C$ iff $A\setminus C \subseteq B\...
Approxiz's user avatar
  • 463
0 votes
0 answers
41 views

How does one prove this inclusion of a set?

I've recently come across the Cauchy's original proof for arithmetic-geometric inequality. In it, he does a proof by induction over a set $A$. Firstly he proves that $P_n\implies P_{2n}$ and then that ...
realreal's user avatar
  • 125
1 vote
1 answer
57 views

Differentiability implies continuity of $f(x,y)$

I have seen proofs of this statement on the site however none use the definition of differentiability I am familiar with which is why I'm asking this question. My proof is as follows: If $f(x,y)$ is ...
Moxy's user avatar
  • 319
1 vote
0 answers
24 views

Constructing a map that accurately mimics 1-point perspective

Problem Provide the set of all possible maps $\Phi(x,y)$ such that the image of the usual square coordinate grid under $\Phi$ is a 1-point perspective grid with vanishing point at $(p,q)$. Note that ...
Simon M's user avatar
  • 875
0 votes
0 answers
33 views

Transcendental nature of natural log for proof validity?

I am following Understanding Analysis by Stephen Abott. I read a well bit into the book but I decided to go through and do the exercises through the book because I felt as if I wasn’t being rigorous. ...
Co-'s user avatar
  • 1
1 vote
2 answers
66 views

Proving $f^{-1}(f(U)) \subset U\ker f$.

Let $f: G \to H$ be a homomorphism between the groups $G$ and $H$. Define $U$ as a subgroup of $G$. I have to prove that $f^{-1}(f(U)) \subset U\ker f$. This is what I found: Take an element $x$ from $...
user33's user avatar
  • 141
0 votes
1 answer
54 views

Wrong proof of Inverse function theorem

Is there anything wrong in my attempt to prove the following section of Inverse function theorem by following: The section I try to prove: Let $f:U\subseteq \mathbb{R}^n\to \mathbb{R}^m$ be ...
HIH's user avatar
  • 449
2 votes
3 answers
61 views

Proof that the supremum of a continuous function is part of the range of that function

I'm following the textbook "Calculus" By Spivak. Currently, I'm reading a proof given for the following theorem: "If $f$ is continuous on $[a,b]$ then there exists an $x^*\in[a,b]$ such ...
Aryaan's user avatar
  • 273
0 votes
0 answers
33 views

Proof of uniform convergence in $ [a,\infty)$ but not $[0, \infty)$

Let $f_n(x)=\frac{x}{(1+x)^n}$. I need to prove that $f(x)=\displaystyle \sum_{n=1}^{\infty}f_n(x)$ is uniformly convergent in $[a,\infty), \forall a>0$ but not uniformly convergent in $[0,\infty)$....
talopl's user avatar
  • 1,009
0 votes
0 answers
33 views

If $f_n(x)\rightrightarrows f(x), ~(f_n(x)-f_n(x\frac{n}{n+1}))n\rightrightarrows g$ then $xf'(x)=g(x)$

I want to prove following statement: Let $f_n:[0,1]\to\mathbb{R}$ be sequence of continuous functions uniformly convergent to $f\left(x\right)$. If $\left(f_n\left(x\right)-f_n\left(x\frac{n}{n+1}\...
Jakub Pawlak's user avatar
0 votes
1 answer
60 views

Change the double integral to an iterated integral

Given: $$\iint_S (x+y)dA$$ such that the bounded region is given by a triangular area with vertices: $(0,0),(0,4),(1,4)$. Now we have that $x = \frac{4}{y}$ and $x =0$ for the first iterated integral ...
samsamradas's user avatar
2 votes
0 answers
53 views

Finding Volume of Revolution Given by $y = \sin x$

The question given is to find the volume of revolution generated by the graph of $y = \sin x$ on the interval $[0, \pi]$. The way I attempted was to form the sums of cylindrical segments given by $\...
Camelot823's user avatar
  • 1,465
0 votes
0 answers
26 views

Number of bounded and unbounded components of nonsingular real algebraic curve $y^2 - p(x)$

I would like to know if someone can kindly verify my solution to Problem 22.2 from MIT's online course on geometry and topology in the plane. Let $C = \{(x,y) \in \mathbb{R}^2 : f(x,y) = 0\}$ be a ...
Menander I's user avatar
0 votes
1 answer
41 views

Proof of radius of convergence

For all $|x|\lt1, \displaystyle\sum_{n=1}^{\infty}a_nx^n$ converges. I need to prove that $\displaystyle \sum_{n=1}^{\infty} a_nx^{2n+1}$ converges for all $|x|\lt1$. Is my proof correct? For all $|x| ...
talopl's user avatar
  • 1,009
0 votes
0 answers
47 views

Prove that for $\forall A \in \mathscr{P}(U) \exists !B \in \mathscr{P}(U) \forall C \in \mathscr{P}(U) $,$C \cap A = C \setminus B$ [closed]

My proof: (exist) $(\to)$ Suppose $ A \in \mathscr{P}(U)$ , $A \in \mathscr{P}(U)$ and $C \cap A$. Letting $B=U\setminus A$, given $x \in C $and $ x\in A $, it follows that $x\in B=U\setminus A$. ...
Darren 's user avatar
1 vote
0 answers
47 views

G a connected Graph. (Dis)Prove the following statement

The number $s(G)$ is the largest natural number $k$ for which there exists a clique $X \subseteq V(G)$ in the graph $G$ with $|X| = k.$ The number $c(G)$ is the smallest natural number $k > 2$ for ...
Barren_Wuffet's user avatar
0 votes
1 answer
86 views

Iterative argument in math proofs

I am reading proof of subspace of $\mathbb{R^n}$ has a basis. And most of them like this(classic proof): let $S\subset\mathbb{R^n}$ ,if $v_1\neq0$ and $<v_1>=S$, we finished proof, otherwise, we ...
MGIO's user avatar
  • 117
3 votes
1 answer
114 views

What is wrong in my solution for the following PDE $u_x^2+u_y^2=1$ with boundary value?

I am asked to solve the boundary value problem on a PDE about $u:\mathbb{R}_{\ge 0}^2\cap \{(x,y): x^2+y^2\ge 1\}\to\mathbb{R}$: $$ \begin{cases} u_x^2+u_y^2=1 & x^2+y^2>1,\, x,y>0\\ u(\cos ...
Asigan's user avatar
  • 1,910
0 votes
1 answer
108 views

number of homomorphism from $K_4\to S_4$ [duplicate]

I want to calculate number of homomorphism from $K_4\to S_4$. let $\phi:K_4\to S_4$ be homomorphism then for each element $x\in S_4$ WHERE $|x|=2$ we have 3 cases , $ord(\phi(a))=1$ and $ord(\phi(b))...
Ricci Ten's user avatar
  • 150
-2 votes
1 answer
91 views

Couldn't Figure Out What is Wrong with My Solution [closed]

My Solution I am currently working on a proof and to express it in a clearer way I used Manim(a math animation software) but for some reason it did not work. So I have to wonder is there anything ...
Secret Mushroom XXX's user avatar
0 votes
1 answer
46 views

Prove that $L(X) \subseteq L(X \cup Y)$.

Given that $L(X)$ is a subspace generated by $X$, any vector in $L(X)$ is a linear combination of the vectors in $X$. That is: $$v=L(X)\rightarrow v=\sum_{i=1}^{m}a_i x_i: (x_i\in X, a_i \in R, \...
Antonius Anonymous's user avatar
0 votes
0 answers
27 views

Proving a sequence of functions isn't uniformly convergent

Let $f_n(x) = x^n(1-x^n)$. I need to prove that the sequence is not uniformly convergent in $[0,1]$. I have already proven that there is a pointwise convergence to $f(x)=0$. However, according to my ...
talopl's user avatar
  • 1,009
1 vote
0 answers
53 views

Altnerative prove to there is a unique $A\in\mathscr{P}(U)$ such that for every $B\in\mathscr{P}(U), A\cup B = B$

Prove that there is a unique $A\in\mathscr{P}(U)$ such that for every $B\in\mathscr{P}(U), A\cup B = B$ I have no trouble proofing its existence. To proof it's Uniqueness. (Letting y and z be an ...
Darren 's user avatar
1 vote
2 answers
80 views

if $f_n \uparrow f$ prove $\mu(f_n\geq t)\rightarrow \mu(f\geq t)$

I think I have to use montone convergence theorem and I have followed this line the sets $f_n\geq t$ are increasing $\lim\mu(f_n\geq t)=\mu(\cup(f_n\geq t)=?\mu(f\geq t)$ if $x\in \cup(f_n\geq t) \...
Dsrksidemath's user avatar
1 vote
1 answer
46 views

Intersection of generalized eigenspaces is trivial (proof validation)

Does the following proof work for showing that the intersection of generalized eigenspaces is trivial? Let $A$ be some $n \times n$ matrix with eigenvalues $\mu,\lambda$ where $\mu\neq \lambda$. The ...
kern711's user avatar
  • 55
2 votes
1 answer
29 views

Probability that before player $X$ throws a $6$, player $Y$ throws what player $X$ threw in the previous throw. Solution verification.

Two players ($X$ and $Y$) take turns rolling the dice. Player $X$ starts the game. Calculate the probability that before player $X$ throws a $6$, player $Y$ throws what player $X$ threw in the ...
thefool's user avatar
  • 1,052
1 vote
1 answer
40 views

A criterion for proving that an algebraic extension $E \subset F$ of fields is normal.

Let $K \subset F$ be fields such that $F$ is an algebraic extension of $K$, if for all elements $\alpha \in F$ there is a field $K \subset E \subset F$ such that $\alpha \in E$ and $E$ is normal over $...
Donlans Donlans's user avatar

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