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

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3
votes
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
votes
1answer
50 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
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
51 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
votes
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? ...
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
64 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
46 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
29 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
145 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
32 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
78 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
104 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} ...
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
36 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
36 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
184 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
21 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
36 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
71 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
32 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 ...
0
votes
2answers
26 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
66 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
44 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
47 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
105 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
155 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 ...
0
votes
2answers
61 views

Proof that $|d(x,y) + d(y,z)| \leq d(x,z)$

Here there is my proof (quite short and easy) of a rather straightforward result. Still, I would like to know: if it is sound, because absolute value always creates me some problem, and if there ...
1
vote
0answers
126 views

Characteristic polynomial of adjoint

I'm trying to show that the adjoint transformation $T^*$ of the endomorphism $T$ on a finite dimensional, real inner product space has the same characteristic polynomial as $T$ in a coordinate free ...
4
votes
0answers
60 views

Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed ...
0
votes
2answers
207 views

Let $f(x) = 1$ for rational numbers $x$, and $f(x)=0$ for irrational numbers. Show $f$ is discontinuous at every $x$ in $\mathbb{R}$

I am working on this proof, and wanted someone to check it and to help me understand what is happening in case (ii). The proof: Let $f(x) = 1$ for rational numbers $x$, and $f(x)=0$ for irrational ...
1
vote
1answer
67 views

Linear maps preserving the determinant and Hermiticity

Conjecture: Let $H_n$ be the space of $n\times n$ complex Hermitian matrices and let $\varphi:H_n \to H_n$ be a linear map which preserves determinants: \begin{equation} \det \circ \varphi = \det. ...
2
votes
5answers
48 views

Let $\{a_n\}$ be a sequence, $a,b$ given so that $\lim\limits_{n\rightarrow\infty} a_n=a$ and $\lim\limits_{n\rightarrow\infty} a_n=b$, then $a=b$.

Let $\{a_n\}$ be a sequence. If $a,b$ given so that $\lim\limits_{n\rightarrow\infty} a_n=a$ and $\lim\limits_{n\rightarrow\infty} a_n=b$, then $a=b$. Proof: Let $\epsilon>0$. Since ...
1
vote
0answers
17 views

Is there other ways to show reverse triangle inequality

I know that for $x,y \in \mathbb{R}$ we have that $$|x-y| \ge ||x|-|y||$$ which can be proven by writing $$|x|=|x+y+(-y)|$$ and $$|y|=|y+x+(-x)|$$ and applying triangle inequality. But I am ...
1
vote
1answer
85 views

Proof that $\lim_{x\to0}\frac{\sin x}x=1$

Is there any way to prove that $$\lim_{x\to0}\frac{\sin x}x=1$$ only multiplying both numerator and denominator by some expression? I know how to find this limit using derivatives, L'Hopital's rule, ...
1
vote
0answers
20 views

Is the following proof correct of why $\gcd(a,b)$ smallest linear combination of $a$ and $b$?

This is the proof I have: Lets see why $\gcd(a, b) $ is the smallest positive linear combination of $a$ and $b$: Let $LC = \{ s'a + t'b : s', t' \in \mathbb{Z}, s'a + t'b > 0 \}$. By the ...
1
vote
2answers
68 views

Is this a correct proof of why $\gcd(a,b) = \gcd(b, a- b)$?

I have a proof but I wasn't sure if it was correct (or how rigorous it is). I will point out what worries me. Let $D_a = \{ d : d \mid a\}$ (i.e. all elements that divide $a$) and similarly $D_b = \{ ...
0
votes
2answers
42 views

How do you show if $d \mid a$ and $d \mid b$ then $d \mid \gcd(a,b)$ without knowing that $\gcd(a,b)$ is a linear combination of $a$ and $b$?

I was trying to prove that if $d \mid a$ and $d \mid b$ then $d \mid \gcd(a,b)$ but wanted a proof that didn't require me to know that $\gcd(a,b) = ax + by$, i.e. that didn't require me to know that ...
2
votes
2answers
62 views

Is there a way to show that $\gcd(a,b) = ax + by $ without also showing that its the smallest positive linear combination?

Is there a way to show that $\gcd(a,b) = ax + by$ without also showing that it is the smallest positive linear combination? i.e. Can it be shown that there exists an $a$ and $b$ such that $\gcd(a,b) = ...
9
votes
2answers
135 views

How to show this cover of $\mathbb{Q}$ doesn't cover $\mathbb{R}$?

Let $\{q_n : n \in \mathbb{N}\}$ be an enumeration of $\mathbb{Q}$ and define $\mathcal{O} = \{I_n : n \in \mathbb{N}\}$ being $$I_n = \left(q_n - \frac{1}{2^n}, q_n + \frac{1}{2^n}\right).$$ It is ...
0
votes
5answers
49 views

Proof of sets. Need an example

My question is to show that $X-(Y \cup Z)$ is a subset of $(X-Y) \cup (X-Z)$. I already did the proof for that and understand that but the second part is to give an example to show that in general, ...
2
votes
1answer
102 views

Starting index of a sequence is irrelevant

"Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers, let $c$ be a real number, and let $k \geq 0$ be a non-negative integer. Show that $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
0
votes
1answer
31 views

Solution Sets of Homogeneous Systems

I had to prove the following theorem: Suppose that $A\mathbf x=\mathbf b$ is consistent for some given $\mathbf b$, and let $\mathbf p$ be a solution. Then the solution set of $A\mathbf x=\mathbf b$ ...