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

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3
votes
1answer
110 views

$\int_A f dm \leq 0 $ for all $A$ lebesgue measurable implies $f \leq 0 $ a.e

$$ \textbf{Problem} $$ $\int_A f dm \leq 0 $ for all $A$ lebesgue measurable set implies $f \leq 0 $ a.e $$ \textbf{Solution (Attempt)} $$ We want to show $X = \{ x : f > 0 \} $ is a null ...
2
votes
2answers
189 views

basic induction probs

Hello guys I have this problem which has been really bugging me. And it goes as follows: Using induction, we want to prove that all human beings have the same hair colour. Let S(n) be the ...
5
votes
0answers
618 views

Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$

As a homework question, we got asked the following: Construct a function $f:[0,1] \rightarrow \mathbb{R}$ which is not the pointwise limit of any sequence of continuous functions Thinking about ...
3
votes
2answers
92 views

What am I doing wrong?

I am trying to prove the integral test for series, but got a strange result. Assume that $f$ is decreasing and positive. Because the series can be imagined as the area-sum of $1$-wide rectangles of ...
2
votes
0answers
139 views

Proving that a matrix $A$ is diagonalizable

I have to prove this result: If $A \in M_n (F)$ has $n$ distinct eigenvalues then $A$ is diagonalizable. My attempt at proof : Let $\lambda_1,\lambda_2,\ldots,\lambda_n$ be the distinct eigenvalues ...
2
votes
0answers
184 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
2
votes
4answers
76 views

Reading, Writing, and Proving Math: Cartesian Product

The following is my attempt at one of my homework assignments. Let A, B, and C be sets. If the statement below is true, prove it. If false, give a counter example. A $\times$ (B $\cap$ C) = ...
1
vote
1answer
202 views

Proof that $\Bbb Z$ has no other subring than itself

I'm asked to prove that the ring of integers $\Bbb Z$ admits no other subring than itself. I'm no too sure about how to prove it. I started using the minimality argument, but I found a ...
0
votes
0answers
107 views

Proving that morphism of sheaves is iso iff induced morphism on stalks is iso

Is the following proof sound/does anyone have another more elegant (categorical) proof? The direction $\Rightarrow$ is obvious the "family of stalks"-functor is a functor and functors take isos to ...
2
votes
2answers
82 views

If $f(x)$ is 2x differentiable in $(a,b)$ & $f'(a)=f'(b)=0$, prove that, $\exists\xi $ in $(a,b)$ S.T. $|f''(\xi )|\leq\frac{4(f(b)-f(a))}{(b-a)^{2}}$

Here is my argument (it doesn't feel 100% correct for some reason): By the mean value theorem, there exists $\xi_{1}$ in $(a,b)$ such that, $$f'(\xi_{1}) = \frac{f(b)-f(a)}{b-a}$$ Since, ...
2
votes
1answer
171 views

Prove that $1/f$ is uniformly continuous on …

I need hints for this particular question: Prove that if a function $f$ is uniformly continuous on $A\subseteq \mathbb{R}$ and $|f(x)|\geq k>0$ for all $x\in A$, then the function $\frac{1}{f(x)}$ ...
1
vote
2answers
106 views

Counter examples to inscribed squares conjecture

Can the counter examples found by me qualify as a counter proof for the "inscribed squares problem" (the Toeplitz' conjecture) ? I ask this here because the problem stands as unsolved for 100 years, ...
4
votes
3answers
189 views

last two digits of $9^{1500}$ (Dummit Foote -Abstract Algebra preliminaries $0.3.5$)

Question is to find last two digits of $9^{1500}$ (No Euler totient theorem please) What i have done so far is : $9^2\equiv 81\text{mod} 100$ $9^4 \equiv 61\text{mod} 100$ $9^8\equiv ...
1
vote
3answers
187 views

If $f(z)$ is continuous inside and on a simple closed contour $C$ and $\int_C f(z)dz=0,$ is $f(z)$ analytic inside $C?$

If $f(z)$ is continuous inside and on a simple closed contour $C$ and $\int_C f(z)dz=0,$ is $f(z)$ analytic inside $C?$ My Attempt: I've found a counterexample: Let $C:|z|=1$ and $f:z\mapsto ...
3
votes
1answer
96 views

Metric spaces and limit points question?

Let $X, d$ be a metric space. For each $x \in X$ and nonvoid $A, B \in X$, define $$d(x, A) = \inf\{d(x, a) : a \in A\}$$ and $$d(A, B) = \inf\{d(a, b) : a \in A, b \in B\}$$ Prove that $d(x, A) = 0$ ...
1
vote
6answers
111 views

Proof of $(\forall x)(x^2+4x+5 \geqslant 0)$

$(\forall x)(x^2+4x+5\geqslant 0)$ universe is $\Re$ I went about it this way $x^2+4x \geqslant -5$ $x(x+4) \geqslant -5$ And then I deduce that if $x$ is positive, then $x(x+4)$ is positive, so ...
2
votes
3answers
224 views

Linear Algebra problem, old Berkeley exam.

I came across this problem: Let $G$ be the group of $2 \times 2$ matrices with determinant $1$ over the four-element field $\bf F$. Let $S$ be the set of lines through the origin in $\bf F^2$. ...
0
votes
1answer
74 views

A question about similar matrices: $Id$ and $-Id$

Currently, I'm trying to understand the idea of matrix similarity. As a toy example, I am thinking about $Id$ and $-Id$. Now, I do not think that these matrices are similar, and here is my proposed ...
0
votes
2answers
74 views

Hint for real analysis question [duplicate]

Suppose that $f$ is one-to-one and continuous on [$a,b$]. Prove that $f$ is either strictly increasing or strictly decreasing on [$a,b$].
2
votes
1answer
44 views

How is this homomorphism from Algebraic Number Theory surjective?

This is on page 72 of Cassels-Fröhlich's Algebraic Number Theory. Take $S$ to be the archimedian valuations of $k$, a finite extension of $\mathbb{Q}$, and define $J_S \subset V_k $ as the subgroup ...
1
vote
2answers
489 views

Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n [duplicate]

Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n Hi everyone, for the proof to the above question, Can I assume that since $(a, b) = ...
2
votes
2answers
58 views

A Better Way to Solve this Factorial Problem?

I had a problem that asked me to find which of the following is larger: ${2013 \choose 500}$ or ${2013 \choose 1500}$ Beneath is my proof. I think it is correct (though your verification and ...
3
votes
1answer
129 views

Please check if my proof is correct of Monotone Convergent theorem

I was required to prove Monotone Convergent Theorem as a corollary of Fatou lemma,i.e using Fatou lemma to prove the MCT. The hint I was given is let $f_n$ be a sequence of increasing function, ...
2
votes
1answer
186 views

Confirming proofs of properties of preimages

I do not think that I made terrible mistakes, nevertheless a conforming word would be good for me. Thank you! Let $f\colon X\to Y$ be a map and $(B_i)_{i\in I}$ a family of subsets of $Y$ ...
-1
votes
1answer
59 views

Showing that a set is open/closed

$\def\R{\mathbb R}$ Is the set $$S=\{(x_1,x_2,x_3) \in \R^3 \mid e^{x_1} + x_2^2 <x_3 \} \subset \R^3$$ open or closed? My attempt: Let $f:\R^3 \to \R$, $f(x_1,x_2,x_3)$ =$e^x_1 + ...
1
vote
1answer
288 views

Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$

Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$ Hi everyone, I would like to know if my assumption is justified for answering the above question. Any ...
3
votes
0answers
361 views

Show that if both a, b are even integers not both 0, then $(a, b) = 2 (a/2, b/2)$

Show that if both a, b are even integers not both 0, then $(a, b) = 2 (a/2, b/2)$. Hi there, I want to know if the following proof I have is strong enough, or if I'm making false assumptions :|. ...
3
votes
1answer
538 views

Finitely generated modules over a Noetherian ring are Noetherian

I'm trying to prove that if the ring $R$ is Noetherian then every finitely generated $R$-module is Noetherian. First of all, it is known that every module is a homomorphic image of a free module, ...
2
votes
1answer
91 views

I just proved that $ℂ$ is not a field. What is the mistake in my reasoning?

What is the mistake in my reasoning? Consider $(X^2+1)$ in $ℝ[X]$. Then $(X^2+1)⊂(X^2,1)$. Because if $f \in (X^2+1)$. Then $f=(X^2+1)g=X^2g+1⋅g$. So $f \in (X^2,1)$. Therefore $(X^2+1)$ not a ...
1
vote
0answers
566 views

Prove Reverse Fatou's lemma

I am trying to prove the reverse Fatou's lemma but I can't seem to get it. Here is my approach: We have a sequence $\lbrace f_k \rbrace$ in $\mathbb R_e$ and $E\subset \mathbb R^n$. We know that ...
2
votes
1answer
60 views

Proof that the Riemann-Integral satisfies $\int_A \lambda f = \lambda \int_A f$

Suppose $A\subset\mathbb{R}^n$ is a closed rectangle and $f:A\to \mathbb{R}$ is Riemann-Integrable on $A$. I want to show that $\lambda f$ is integrable and that $$\int_A \lambda f =\lambda\int_Af $$ ...
1
vote
1answer
42 views

Help with understanding this proof (I think it's Hensels Lifting?)

I am reading a proof that shows that if $a$ is a quadratic residue modulo $p^k$, where $p$ is a prime > $2$ and $k$ is a positive integer, then it is also a quadratic residue modulo $p^{k+1}$. Here ...
1
vote
1answer
102 views

Supremum of sets of extended reals

Hi everyone I'd like to know if following is really correct, looks kinda cumbersome, I think, it is for the great quantity of cases to analyze. To be honest I don't know if this is the better way to ...
0
votes
0answers
36 views

Set theory problem on existence of functions

May I verify if my proof to the b/m claim is correct? Thank you! Let $\mathbb{N}^{+}= \mathbb{N}-\{0\}.$ Suppose $h: \mathbb{N} \times \mathbb{N} \to \mathbb{N}^+$ is a function. For each $m\in ...
3
votes
2answers
60 views

Proof of statement: If $a\mid b$ and $a\mid c$, then $a \mid b+c$

Statement: If $a$ divides both $b$ and $c$, then $a$ divides $b+c$ Proof: Assume that $a$ does not divide $b+c$. Then there is no integer $k$ such that $ak=b+c$. However, $a$ divides $b$, so $am=b$ ...
3
votes
1answer
153 views

Proof with quantifiers

$(\forall x)(\exists y)(x+y=0)$ $x$ and $y$ are real numbers The statement reads: for all $x$ there exists some $y$ such that $x+y=0$ is true. My proof is: take $y=-x$ Is this valid? I'm just ...
1
vote
1answer
123 views

Let $p,q$ be distinct primes. Find number of generators of $(\mathbb{Z}/pq\mathbb Z, +)$

May I verify if my proof to the a/m claim is correct? Thank you. #generators $=\phi(pq)$. Let $ A = \{x\in \mathbb{N}: q\mid x \wedge x< pq\}$ and $B = \{y\in \mathbb{N}: p\mid y \wedge y< ...
3
votes
1answer
87 views

For bijection $f:A \rightarrow B$, prove that $f^{-1} \circ f = {\text {id}}_{A}$

I have to prove that for a bijection $f:A \rightarrow B$, $f^{-1} \circ f = {\text {id}}_{A}$, where ${\text {id}_A}$ is the identity function of $A$, and we define $f^{-1}: B \rightarrow A$ by ...
1
vote
1answer
970 views

Is it logically valid to prove DeMorgan's laws using the duality of boolean algebra?

I'm taking an introductory course in boolean algebra, and have been assigned the task of proving DeMorgan's Laws (so, disclaimer, this is homework). One line of reasoning that I came up with would be ...
0
votes
2answers
128 views

proof that the three interior angles of a triangle is congruent to a straight line (without measurements)

I'm trying to essentially prove that the interior angles of a triangle add up to 180 degrees. However, I'm trying to prove it without mentioning measurements of an angle. I think I understand the ...
3
votes
4answers
138 views

Show that $\int_{1}^\infty \dotsb\int_{1}^\infty \frac{dx_1 \dotsb dx_n}{x_1^{\alpha_1}+\dotsb + x_n^{\alpha_n}}<\infty$

Here's my solution to an old qualifier problem. Would you tackle it differently? Is there a flaw in my work? Suppose that $\alpha_1, \dotsc, \alpha_n$ are positive numbers such that ...
1
vote
2answers
41 views

Verify that the transform of $y(t) = t^2e^{at}$ is $Y(s) = \frac{2}{(s-a)^3}$

I made the distinction to amplify "=" 3 times for easier readability. I tried: $$F(s) === \int_0^\infty t^2e^{(a-s)t}dt === \frac{1}{a-s}e^{(a-s)t}t^2\Big|_0^\infty \ - \frac{2}{a-s}\int_0^\infty ...
2
votes
1answer
137 views

How to show that time-dependent norm is continuous (please verify my proof)

For each $t \in [0,T]$, let $H_t$ be a Hilbert space. Suppose for each $t$, the operator $T_t:H_0 \to H_t$ is a linear homeomorphism with inverse $T_{-t}:H_t \to H_0$ also linear homeomorphism. ...
0
votes
0answers
39 views

$G =\{z\in \mathbb{C}: z^n =1\}\leq (\mathbb{C} -\{0\}, \times) \wedge \exists g\in G: |G| = |\left \langle {g} \right \rangle| = n. $

may I verify if my proof to the a/m claim is correct? Thank you. Lemma: Let $G$ be a finite group. $|\{x \in G: x^m =e\}|\leq m$ $\Longleftrightarrow G$ is cyclic. Proof: Suppose $|\{x \in G: x^m ...
2
votes
1answer
78 views

Let $f: G \to H$ be a homomorphism and $g \in G. $ If $o(g)=n$ is finite, then $o(f(g)) | o(g).$ If $f$ is an isomorphism, then $o(f(g)) =o(g)$

May I verify if my proof to the a/m claim is correct? Thank you. Let $e_G, e_H$ be identity elements of $(G,*_1)$ and $(H,*_2)$. Note that $f(e_G)*_2f(x) = f(x) , \forall x\in G.$ Then $e_H*_2f(x) ...
2
votes
0answers
33 views

$(\mathbb{Z}/(p) - \{[0]\}, \times)$ is a group $\Longleftrightarrow p$ is a prime.

May I verify if my proof to the a/m claim is correct? Thank you. Suppose $G = (\mathbb{Z}/(p) - \{[0]\}, \times)$ is a group. Given $[a] \in G, $ where $0 < a \leq p-1,$ $\exists! [b] \in G$ such ...
5
votes
3answers
2k views

Prove that $x^3 + x^2 = 1$ has no rational solutions?

Is this enough for a proof?: $$x^3+x^2 = 1$$ I would factor and get: $x^2(x+1) = 1$ I would show that $x = \sqrt1$, which is irrational but then do I have to show more? $x+1=1$ which gives me $x=0$ ...
1
vote
1answer
116 views

Injectivity and Imdepotency implies Surjectivity

This question stem from Natural Deduction (FeedBack). The reason why I think it is justifiable to open this up as a separate question is that I am now considering other measures to show it, possibly ...
4
votes
1answer
110 views

Function Surjectivity Proof

I have this question: Prove that a function $f:X\rightarrow Y$ is surjective iff for any finite set $Z$ and any function $g:Z\rightarrow Y$ there exists a function $h:Z\rightarrow X$ such that ...
1
vote
1answer
54 views

Introduction to Analysis: Properities of Functions

If I remember correction from my abstract algebra course, if $f(x)$ is defined for all x and is bounded, then composition mapping $f\cdot g$ is also bounded, and so should $g\cdot f$ since the range ...