If you already have a proof for some result, but want to ask for a different proof (using different methods).

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1answer
69 views

For all $x$, $f(x)=\int_{0}^{x}f(t)dt$. Prove that $f(x)=0$ for all $x$.

Suppose that the function $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and that $$f(x)=\int_{0}^{x}f(t)dt\qquad\text{for all $x$}$$ Prove that $f(x)=0$ for all $x$. Attempt Since ...
4
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0answers
50 views

Monomorphisms of monoids are stable under coproducts

Let $M,N,K$ be three monoids (or even groups, if you like) and let $N \to K$ be an injective homomorphism. Then, the induced morphism $M \sqcup N \to M \sqcup K$ is also injective. This is easy to ...
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2answers
68 views

Alternative proof: G group of order $p^2$, p prime $\Rightarrow$ $H$ is normal in $G$.

Let $G$ be a group of order $p^2$ where $p$ is prime. If $H$ is a subgroup of order $p$, show that $H$ is normal in $G$. I would like to prove this with the tools that the book has provided up to ...
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1answer
20 views

On a Simple Theorem from Hilbert's *The Foundations of Geometry*

I want my proof writing skills to get better. I am trying to do this through proving theorems from Hilbert's axioms for Euclidean Geometry. I found Hilbert's The Foundations of Geometry here, a ...
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2answers
38 views

Is there a name for this double summation identity? What is the shortest way to illustrate that it holds?

Say I have the following the expression: $$\sum\limits_{j=0}^{i-1} \sum\limits_{u=0}^{j} g(u)$$ By enumeration, it is easy to see that: in the case when $j=0$ we have $$\sum\limits_{u=0}^{j} g(u) ...
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0answers
31 views

Use Cauchy-Schwarz inequality to prove that $\langle\,,\rangle : \mathscr H \times \mathscr H \to \Bbb C$ is continuous.

Let $(a,b) \in \mathscr H \times \mathscr H$ be fixed. So we have to prove that for a given $\epsilon \gt 0$, we can find $\delta_1 \gt 0$ and $\delta_2 \gt 0$ such that $\lvert \langle x,y\rangle - ...
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0answers
46 views

Are 2 quadrilaterals similar if they are both inscribed and have congruent angles and have perp diagonals

This is problem 365 from Kiselev's Planimetry book. I have to show that two inscribed quadrilaterals with perpendicular diagonals are similar iff they have respectively congruent angles. Here is my ...
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2answers
49 views

Can one show that $\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||}$ is always on the range of $\cos \theta $?

A basic property of the dot product of two vectors is that $$\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||} = \cos \theta $$ Where $\theta$ is the angle between the two vectors. Since there is ...
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1answer
56 views

Prove $T$ is invertible

If $T\in L(X,X)$ where $X$ is a Banach space and $L(X,X)$ is denoted as the space of bounded linear maps, and $\|I-T\|<1$ where $I$ is the identity operator, then $T$ is invertible? Here ...
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0answers
23 views

Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
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0answers
38 views

Solving a Diophantine Equation with 2 variables

This is my answer for the following question: Find all natural numbers $(a,b)$ for which $a^b-b^a=1$. When $a$ or $b$ equals $1$, $(a,b)=(2,1)$ is trivial. If $a,b>1$, I generalized the problem ...
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0answers
32 views

Direct proof of the existence of optimal memoryless deterministic policies in MDP

It is well known that (finite-state, finite-action, discrete time) MDPs admit an optimal policy that is memoryless and deterministic (sometimes called pure). The proof of this fact for ...
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0answers
35 views

Proof $f(x,y)=x_1+e^{x_{2}}$ is strictly convex

I am trying to show that $f(x,y)=x_1+e^{x_2}$ is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ...
4
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3answers
86 views

Proving that $\sqrt{a_1^2} + \sqrt{a_2^2} +…+ \sqrt{a_n^2} > \sqrt{a_1^2 + a_2^2 +…+a_n^2}$ using Pythagoras

I think I have a proof using Pythagoras for $\sqrt{a_1^2} + \sqrt{a_2^2} > \sqrt{a_1^2 + a_2^2}$. I'm interested in whether there's a way to use that proof with Pythagoras to prove the general ...
2
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1answer
38 views

Generalisation of Binomial Theorem, Leibniz Formula and similar theorems [duplicate]

Since the beginning of the year, our maths teacher showed us the Binomial Theorem in $\mathbb{R}$\, then in $\mathbb{C}$\, in $M_n(\mathbb{K)}$ with two matrices which commute, and now the Leibniz ...
3
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2answers
158 views

Proving finite additivity for this semi-algebra (infinite coin flips)

Background copied and pasted from another one of my questions: Background: Consider flipping a coin $n$ times. Define the sample space as $$ \Omega = \{(r_1,r_2,r_3,\dots); r_i = 0 \text{ or }1\} $$ ...
2
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2answers
71 views

Dummit and Foote exercise verification?

I was working on the following problem: Let $\sigma$ be the m-cycle $(1 2...m)$. Show that $\sigma^{i}$ is also an m-cycle iff $\gcd(i,m)=1$ A solution to this problem is given here. But the ...
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1answer
43 views

Verify proof that if $M,N$ are $R$-modules and $M$ is Noetherian, $N$ is finitely generated, then $M\otimes_R N$ is Noetherian

I have to prove that If $M,N$ are $R$-modules and $M$ is Noetherian, $N$ is finitely generated, then $M\otimes_R N$ is Noetherian We let $S$ be a non-finitely generated submodule of $M\otimes_R ...
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3answers
191 views

Isomorphism of Non-Symmetric Matrices

$A, B$ are non-symmetric matrices of dimension $m \times n$ where $m=n$ or $m \neq n$. Example: An example of $6 \times 3$ non-symmetric matrix is $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & ...
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4answers
113 views

Alternative way to show that a simple group of order $60$ can not have a cyclic subgroup of order $6$

Suppose $G$ is a simple group of order $60$, show that $G$ can not have a subgroup isomorphic to $ \frac {\bf Z}{6 \bf Z}$. Of course, one way to do this is to note that only simple group of ...
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2answers
77 views

Proof for $ \frac{2}{\pi}x \lt \sin{x} $ for $ x \in (0,\frac{\pi}2) $

The following is part of exercise 6.26.21 from Tom Apostol's Calculus Volume 1. I wonder if my proof is correct and if there is a simpler alternative proof. Prove the following by examining the ...
4
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1answer
93 views

A Scalar times the Zero Vector

I'm reading Linear Algebra Done Right by Sheldon Axler and the proof given in the book is the same as the one in the answer provided for this question. I tried to solve this before looking at the ...
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5answers
134 views

Better proof for $\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$

It's required to prove that $$\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$$ I managed to go about out it two ways: Assume it holds: $$\frac{1+\cos x + \sin x}{1 - ...
4
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0answers
110 views

A real symmetric matrix $A$ positive definite if all its eigenvalues are positive

Let $A\in \mathbb R^{n \times n},\ A^T=A$ and the eigenvalues $\lambda_i>0$. Then $v^TAv>0$ for every nonzero vector $v$. I know how to prove the above statement by using the fact that if ...
3
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1answer
53 views

Verify proof of $\sum_{k=1}^{n}{k^{-1/2}}<2\sqrt{n}$ for every $n\ge1$

The following is exercise 2.6.20 (c) from Tom Apostol's Calculus Volume 1, I'd like someone to verify my proof. I'm also interested in simpler alternative proofs. Determine if the following is ...
4
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1answer
101 views

Prove that $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$

Theorem Let $\{a_n\}$ and $\{b_n\}$ be convergent real sequences. Assume that there exists a $N\in\mathbb{N}$ so $a_n\le b_n$ (eq. 1) for all $n\ge N$. Then $\lim_{n\to\infty}a_n\le ...
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2answers
104 views

How do i evaluate this sum :$\sum _{m=1}^{\infty } \sum _{k=1}^{\infty } \frac{m(-1)^m(-1)^k\log(m+k)}{(m+k)^3}$?

How do I evaluate the following sum: $$\sum _{m=1}^{\infty } \sum _{k=1}^{\infty } \frac{m(-1)^m(-1)^k\log(m+k)}{(m+k)^3}$$ Note I used many idea such as :Hochino's Idea and taylor expansion of ...
4
votes
2answers
67 views

Tangent identity given $a + b + c = \pi$

Given that $a + b + c = \pi$, that is, three angles in a triangle - then prove that $$\tan a + \tan b + \tan c = \tan a \tan b \tan c$$ Is my solution below completely rigorous? Can I justify taking ...
6
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1answer
159 views

How to Prove the Chain Rule for Limits Using a $\varepsilon-\delta$ Argument?

I came across the chain rule for limits the other day and it interested me quite a bit and surprisingly I couldn't find the proof on the internet anywhere. From what I understand the chain rule for ...
3
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2answers
78 views

Prove that if $G$ is a group of order $39$ then $G$ has a subgroup of order $3$

I was able to show this by first proving $G$ requires and element of order $3$. However I am looking for alternative proofs without the use of Sylow theorems or Cauchy's theorem. Any hints would be ...
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2answers
54 views

Are topologies on $\Bbb R$ with bases $\{[-n,n]:n\in\Bbb N\}$ and $\{(-n,n):n\in\Bbb N\}$ homeomorphic?

I think that NO because there is no way to map an open set of the kind $[-n_1,n_1]$ to some open map of the kind $(-n_2,n_2)$. Proof: imagine some homeomorphism $f:(\Bbb R, T_1) \to (\Bbb R, T_2)$ ...
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0answers
46 views

What is $\mathbb Q +A=\{q+a : q\in \mathbb Q , a\in A\subset [0,1]\}$ $?$ [duplicate]

What is $$\mathbb Q +A=\{q+a : q\in \mathbb Q , a\in A\}$$ where $A$ is a subset of the interval $[0,1]$ with non-empty interior $?$ $A.\mathbb Q+A=\mathbb R$ $B.\mathbb Q+A $ can be a ...
1
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1answer
76 views

Prove that if a group $G$ has $|G| = 6$ then $G$ is isomorphic to either $\Bbb Z/6$ or $S_3$

Prove that if a group $G$ has $|G|$ = 6 then G is isomorphic to either $\mathbb{Z}_6$ or $S_3$. I have the proof by contradiction but I was wondering if there was a direct proof instead in case I ...
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1answer
47 views

Proving that the preimage of an open set is open

I am trying to learn how to prove that the preimage of an open set is open in general topology. Here is an example that I am not really satisfied with Proposition 3.9 (Book: Essential Topology, ...
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1answer
47 views

Frattini subgroup of a finite elementary abelian $p$-group is trivial

I would like to improve my proof of the following result: If $H$ is a finite, elementary abelian $p$-group, then $\Phi(H) = 1$. Here, $\Phi(H)$ is the Frattini subgroup, defined as the ...
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1answer
42 views

Prove that a group of order 25 has a subgroup of order 5

I found this question Subgroup(s) of a group of order 25 I want to know if proving such a statement is possible by contradiction. Question: Let G be a group of order 25. Prove that G has at least ...
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1answer
46 views

Verify proof of $ f(x)=e^x $ if $ f(x+y)=f(x)f(y) $ and $ f'(x)$ exists for all $x$

This is exercise 6.26.8 from Tom Apostol's Calculus I, I'd like to ask someone to verify my proof. I'd be also interested in alternative proofs: If $ f(x+y)=f(x)f(y) $ for all $ x $ and $ y $ and if ...
2
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1answer
68 views

Triple Products are Isomorphic

I am currently working through Awodey's Introduction to Category Theory, and I'm learning how to move around complicated diagrams. I want to show that $A\times(B\times C)\cong(A\times B)\times C$; ...
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2answers
33 views

Condition for inverse of quadratic function - alternative solutions

I was helping my friend teacher to assemble a list of exercises to their precalculus students. So I came up with this problem: Let $f$ be a quadratic function, i.e. $$f(x) = ax^2 + bx + c,$$ ...
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1answer
88 views

Is this a sufficient proof of a math contest problem?

Problem: If a,b,c,d are real, prove that $$a^2+b^2=2$$ $$c^2+d^2=2$$ $$ac=bd$$ Is true if and only if $$a^2+c^2=2$$ $$b^2+d^2=2$$ $$ab=cd$$ My proof is as follows: Note that each of the ...
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1answer
30 views

Is there a shorter proof to show that this complex intergral is constant?

I have the integral, $$I(R) = \int_{C_R}\frac{1}{z(z-1)^2} dz$$ with the property that $$\left|\frac{1}{z(z-1)^2}\right| \leq \frac{1}{R(R-1)^2} \quad |z|=R>1$$ Where $C_r$ is the contour ...
2
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1answer
63 views

If $\limsup_{n\to\infty} \ x_{n} = a$ then why does it exist a subsquence $s_{n}$, which $\lim_{n\to\infty} \ s_{n} = a$?

Maybe it seems trivial since $\limsup$ is known as the "greatest limit point of $x_n$", so there's a subsequence which converges to $a$. But I cannot use this definition. Is it possible to prove it ...
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4answers
81 views

n-cents stamp (Strong induction)

Imagine that your country's postal system only issues 2 cent and 5 cent stamps. Prove that it possible to pay for postage using only these stamps for any amount n cents, where n is at least 4. My ...
3
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3answers
59 views

Show that $\int_1^\infty \frac{\ln x}{\left(1+x^2\right)^\lambda}\mathrm dx$ is convergent only for $\lambda > \frac{1}{2}$

Show that the improper integral $$\int_1^\infty \frac{\ln x}{\left(1+x^2\right)^\lambda}\mathrm dx$$ is convergent only for $\lambda > \frac{1}{2}$. We will show that the sequence of integrals ...
0
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1answer
49 views

Prove that a strictly increasing function $f:[a,b]\rightarrow\mathbb{R}$ which has the intermediate value property is coninuous on $[a,b]$. [duplicate]

Prove that a strictly increasing function $f:[a,b]\rightarrow\mathbb{R}$ which has the intermediate value property is continuous on $[a,b]$. Let $x_0\in[a,b]$. As $f$ is strictly increasing, ...
4
votes
4answers
88 views

Show that $\lim\limits_{x\rightarrow 0}f(x)=1$

Suppose a function $f:(-a,a)-\{0\}\rightarrow(0,\infty)$ satisfies $\lim\limits_{x\rightarrow 0}\left(f(x)+\frac{1}{f(x)}\right)=2$. Show that $$\lim\limits_{x\rightarrow 0}f(x)=1$$ Let ...
0
votes
1answer
28 views

Is it possible to prove by contradiction that the boundary of a set in a metric space is closed using these definitions.

The definitions given are the following: Given a metric space $(X,d)$ A set $C \subset X $ is open iff for every $c \in C$ there exist an open ball $B(c,r) \subset C$. Where $r$ is the radius of ...
1
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0answers
27 views

Is there a proof for area theorem, which does not use area argument?

Area Theorem Let $f(z)=z+b_0 + \frac{b_1}{z} + \frac{b_2}{z^2} + ... $ be an injective holomorphic function defined in the domain $|z|>1$. Then, $\sum_{n=1}^\infty n|b_n|^2 \leq 1 $. ...
1
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2answers
49 views

Non-integral-over-a-point proof that the probability of any point in a continuous distribution is zero

My Question For continuous random variables / continuous distributions, it is defined that the probability of any point has probability $0$. The most common proof for this is as follows: ...
0
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1answer
108 views

$X$,$Y$,$Z$ mutually independent implies $X+Y$ independent of $Z$

Supposing $X$, $Y$ and $Z$ and mutually independent real random variables, how can we prove that $X+Y$ and $Z$ are independent from the definition? If not from the definition, using $\sigma$-algebras? ...