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

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4
votes
2answers
73 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
votes
1answer
168 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
votes
2answers
96 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 ...
1
vote
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 $f:(\Bbb R, T_1) \to (\Bbb R, T_2)$ ...
2
votes
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
vote
1answer
78 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 ...
0
votes
1answer
50 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, ...
1
vote
1answer
58 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 ...
1
vote
1answer
43 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 ...
3
votes
1answer
49 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
votes
1answer
70 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 $A\times(B\times C)\cong(A\times B)\times C$; ...
3
votes
2answers
36 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. $$f(x) = ax^2 + bx + c,$$ ...
2
votes
1answer
89 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 ...
1
vote
1answer
33 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
votes
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 ...
1
vote
4answers
89 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
votes
2answers
67 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 on ...
0
votes
1answer
55 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 $\epsilon&...
0
votes
1answer
29 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
vote
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
vote
2answers
60 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: $$\Pr(X=a)=\...
0
votes
1answer
123 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? ...
1
vote
1answer
37 views

How would you write formal definition for indeterminate limit?

The original one, I believe, should be that for $$\lim_{{x}\to{\infty}}f(x)=L$$ $\forall\epsilon>0, \exists M \in ℝ$ such that $x>M \Rightarrow |f(x)-L|<\epsilon$ But what if it is that x ...
0
votes
2answers
65 views

Circular argument in proof?

See the part (B). In it, the author "proves" the limit $\lim\limits_{|x|\to\infty}\left(1+\frac 1x\right)^x$. The part concerning $x\to -\infty$ is in the next page but in that, he just takes $y=-x$ ...
0
votes
1answer
55 views

Proving that $ \chi(G) = \omega(G) $ if $ \bar{G} $ is bipartite.

I know that $ \chi \! \left( \bar{G} \right) = 2 $ and that $ \chi(G) \geq \alpha \! \left( \bar{G} \right) $, but how can I conclude that $ \chi(G) = \omega(G) $?
0
votes
0answers
30 views

Proving König-Egerváry by induction in the number of vertices

I was assigned to an exercise I've been struggling with. Let an essential vertex be a vertex that belongs to every maximum matching of a graph G. Using that, prove that, for every bipartite graph, ...
7
votes
4answers
148 views

A new approach to find value of $x^2+\frac{1}{x^2}$

When I was teaching in college class ,I write this question on board . if we now $x+\frac{1}{x}=4$ show the value of $x^2+\frac{1}{x^2}=14$ Some student ask me for multi idea to show or prove that ...
1
vote
1answer
33 views

Proof strategy about a property of triangular matrices

Is it by mathematical induction the best way to prove that the determinant of an upper (lower) triangular matrix is the product of the elements of the main diagonal? Actually, I am wondering about ...
6
votes
0answers
87 views

Elementary proof that finite sums of square roots of primes is irrational

It is relatively easy to show that if $p_1$, $p_2$ and $p_3$ are distinct primes then $\sqrt{p_1}+\sqrt{p_2}$ and $\sqrt{p_1}+\sqrt{p_2}+\sqrt{p_3}$ are irrational, but the only proof I can find that $...
4
votes
7answers
105 views

Prove $ \forall x >0, \quad \sqrt{x +2} - \sqrt{x +1} \neq \sqrt{x +1}-\sqrt{x}$

I would like to prove $$ \forall x >0, \quad \sqrt{x +2} - \sqrt{x +1} \neq \sqrt{x +1}-\sqrt{x}$$ I'm interested in more ways of proving it My thoughts: \begin{align} \sqrt{x+2}-\...
0
votes
1answer
40 views

Every function $f:\mathbb{N}\rightarrow\mathbb{R}$ is continuous (using definition)

The standard proofs can be found here: Every function $f: \mathbb{N} \to \mathbb{R}$ is continuous? But I want to see how this could be proved using directly the definition of continuity of real ...
1
vote
1answer
39 views

Is $H$ the Commutator Subgroup of $G$ if $H\neq \{1\}, H\leq G$ and $[H, G] = [H, H]$?

I'm working on the following problem obtained when trying to prove in a different way that the alternating group $A_n$ is the commutator subgroup of the symmetric group $S_n$ if $n\geq 5$. (Again, I ...
0
votes
1answer
37 views

Prove that for any polynomial $P(x)= a_nx^n + \cdots +a_1x+a_0,P$ is differentiable

Prove that for any polynomial $P(x)= a_nx^n + \cdots +a_1x+a_0,P$ is differentiable, and $P'(x) = na_nx^{n-1}+\cdots+2a_2x+a_1.$ I am trying to figure out a way to prove this with out having to use ...
2
votes
0answers
193 views

Assumptions needed for proof of the Pythagorean Theorem from examples

There are lots of geometric dissection and reassembling proofs of the Pythagorean Theorem (PT). I want to know what are the necessary ideas to allow a proof using discrete instances. For example, we ...
1
vote
1answer
31 views

How to show that $x_n = \sum_{k=1}^{n} \frac{\cos(k+1)x - \cos kx}{k}$ converges using Cauchy convergence theorem?

I asked a question. It is solved by using Dirichlet's test. Is it possible to show that $x_n = \sum_{k=1}^{n} \frac{\cos(k+1)x - \cos kx}{k}$ converges using Cauchy convergence theorem? Thank you ...
0
votes
0answers
23 views

Another proofs of the comparison principle for PDEs with nonlocal term (derivative)

In my research I study (fully) nonlinear PDEs with fractional derivatives. For an analysis of these, I use the theory of viscosity solutions. So I am now at the end of my tether becasuse I can not ...
1
vote
1answer
41 views

Overlapping of unit length real lines?

X, Y and Z are closed intervals of unit length on the real line. The overlap of X and Y is half a unit. The overlap of Y and Z is also half a unit. Let the overlap of X and Z be k units. Which of the ...
2
votes
0answers
45 views

Shouldn't this be the standard way to prove that if $p>1$ then $p^n \to +\infty$?

Proposition: Let $p>1$. We have $$ \lim_{n\to +\infty} p^n = +\infty.$$ My professor proved this setting $p=1+\varepsilon$ for some $\varepsilon>0$ and then using Bernoulli's inequality and the ...
1
vote
2answers
74 views

How to complete the following proof of Bolzano's Theorem?

I am trying to prove Bolzano's Theorem using the following argument. But the problem actually is that though "intuitively" I can "see" why the argument works (if I am not wrong in "seeing"), I can't ...
1
vote
2answers
76 views

Is an isometry necessarily surjective?

A mapping $f:X\to Y$ between metric spaces $(X,d_X)$ and $(Y,d_Y)$ is called an isometry if it preserves distances, i.e. $$d_Y(f(a), f(b))=d_X(a,b)\text{ }\forall\text{ } a,b\in X$$ My question is: ...
2
votes
2answers
34 views

$\bigcup\limits_{i=1}^n A_i$ has finite diameter for each finite $A_i$

Let $(X,d)$ be a metric space. The diameter of a set $A\subset X$ is defined to be $\operatorname{diam}(A)= \sup\{d(x,y):x,y\in A\}$. Suppose $A_1, \dots, A_n$ is a finite collection of subsets of $X$...
0
votes
2answers
29 views

Check my proof of showing that diam$(A)=$ diam$(\bar{A})$

Let $(X,d)$ be a metric space. The diameter of a set $A\subset X$ is defined to be diam$(A)=$ sup$\{d(x,y):x,y\in A\}$. Show that for any set $A\subset X$, diam$(A)=$ diam$(\bar{A})$ where $\bar{A}$ ...
2
votes
2answers
74 views

What is the use of Dirichlet Integral? [closed]

How can I find the value of $$\large\int_0^\infty\left(\dfrac{\sin x}x\right)^5dx$$ using Contour Integrals? I attempted it using Integration by Parts and got the an got the answer. I have studied ...
5
votes
0answers
48 views

Complex 'mean-value-theorem'-like property implies quadratic

One of my friend asked me the following problem: Problem. Suppose that $f$ is a holomorphic function on a convex open set $U$ which satisfies the following property: For all distinct $z, w \in U$, ...
1
vote
0answers
58 views

Is Doobs theorem of binary rank really true?

The theorem states that any adjacent matrix of the line graph of a connected graph has a binary rank n-1 if the order, n, of the graph is odd. I have pondered about this and found that it doesn't ...
2
votes
2answers
137 views

If all continuous functions $f: X\subset \mathbb{R}\to \mathbb{R}$ are bounded then $X$ is compact

I'm trying to show that in $\mathbb{R}$ a pseudocompact set is compact. That is, if $X\subset \mathbb{R}$ is such that all continuous functions $f: X\to \mathbb{R}$ are bounded, then $X$ is compact. ...
3
votes
0answers
68 views

If $\liminf\, |a_n|=0.$ Does there exists a subsequence of $\{a_n\}$ which has finite sum? [duplicate]

If $\liminf\, |a_n|=0.$ Does there exists a subsequence of $\{a_n\}$ which has finite sum? I tried to prove as follows: Since $\liminf\, |a_n|=0,$ then we can find $n_1<n_2<n_3\ldots$ such ...
1
vote
1answer
206 views

The quotient of a Dedekind domain by a principal ideal is a principal ideal ring.

Let $A$ be a Dedekind domain, and $a\in A-\{0\}$. I have to prove that every ideal of $A/(a)$ is principal. This is a particular case of the exercise 9.7 in Atiyah's Introduction to Commutative ...
0
votes
2answers
51 views

Proof of the Reverse Triangle Inequality

Here there is my proof (quite short and easy) of a rather straightforward result. The text of this question comes from a previous question of mine, where I ended up working on a wrong inequality. Here ...