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

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5
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1answer
181 views

Proof of Faà di Bruno's formula using a convolution identity for Bell polynomials?

I have noticed there is an identity for Bell polynomials that can apply of Faà di Bruno's formula. This is a convolution identity that states: $$ (x \ast y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j ...
16
votes
9answers
365 views

Lesser known derivations of well-known formulas and theorems

What are some lesser known derivations of well-known formulas and theorems? I ask because I recently found a new way to derive the quadratic formula which didn't involve completing the square as is ...
0
votes
0answers
22 views

Linear independency question in $\mathbb{R}^3$ and cross product

If $u$ and $v$ are non-parallel vectors $\in \mathbb R^3$, is it true that the vectors $(u+v,u-v,u\times v)$ form a basis of $\mathbb{R}^3$? My answer would be affirmative supposing that $u$ ...
1
vote
4answers
51 views

Alternative proof for $\forall n\in\mathbb N (n\ge 6 \Rightarrow \exists a,b\in\mathbb N : n=3a+4b)$

Can I use only strong induction in order to prove $$\forall n\in\mathbb N (n\ge 6 \Rightarrow \exists a,b\in\mathbb N : n=3a+4b)$$ Is there any other option?
1
vote
1answer
93 views

Use division algorithm to prove for any odd integer n, $n^2 -1$ is a multiple of 8.

Here is what I know if n is any odd integer then $n$ can be expressed as $n=2k+1 ~~~ where~k\in\mathbb{Z}$.So $n^2-1=(2k+1)^2 -1=4k^2+4k=4k(k+1)$ but $k(k+1)~~ is~~even$. Thus $k(k+1)=2t, t\in ...
0
votes
1answer
84 views

Proving Euler Summation by Parts Without Using Integration by Parts

Assume $f$ has continuous derivative $f'$ on [a,b]. Prove the following summation formula, without using partial integration: \begin{equation} \sum_{a< x \le ...
1
vote
0answers
30 views

Riemann-Stieltjes Integrals with $n$ discontinuities (Proof Review)

First of all is the proof legitimate? If so, then are there other methods to achieve the same result that are neater than mine? Let $\alpha : [a,b] \to \mathbb{R}$ be a step function with ...
1
vote
1answer
76 views

Special Integral Proof

How to prove $$\int_0^\infty x^{2n-1} \exp(-a^{x^3})\, dx = \frac{\Gamma(n)}{2a^n} ,\quad n> 0 ,\quad a>0. $$
1
vote
1answer
45 views

Question on series $\frac {\Gamma'(z)}{\Gamma(z)}$

Prove that: $$\frac {2\Gamma'(2z)}{\Gamma(2z)}-\frac {\Gamma'(z)}{\Gamma(z)}-\frac {\Gamma \prime(z+\frac{1}{2})}{\Gamma(z+\frac{1}{2})} =2 \log 2$$ But I obtain this equal zero: $$\frac ...
0
votes
1answer
43 views

Need alternative proof to $ \exists x (k(x) \rightarrow t)$ entails $\forall x (k(x)) \rightarrow t $

I tried to prove $ \exists x (k(x) \rightarrow t)$ entails $\forall x (k(x)) \rightarrow t $ as; $ \exists x (k(x) \rightarrow t)$ $ \exists x (\neg k(x) \lor t)$ $ \exists x (\neg k(x)) \lor ...
0
votes
1answer
45 views

Elements of order 5 in $A_6$

I am trying to find the elements of order 5 in $A_6$ and I understand that they are of the form $(abcde)$, correct? So the number of elements is $(6*5*4*3*2)/5$=144. I looked somewhere else and it ...
0
votes
1answer
51 views

Placing bricks on Board

Situation: I have a $8\times 8$ board (sic), but two squares from it's one diagonal are removed (Black colored squares are removed) I'm given with plenty of(Rectangular) bricks having dimensions ...
0
votes
2answers
97 views

Prove that every nonempty finite set has a maximum.

how do I prove that every nonempty finite set has a maximum. I know how to explain this by words but didn't know how to put it into mathematical form. I found a way to prove this by induction in ...
3
votes
3answers
91 views

$\frac{1}{1+2}+\frac{1}{1+2+3}+\dots+\frac{1}{1+2+3+\dots+x}=\frac{2011}{2013}$

I want to see OTHER approaches than this one. Make sure they are significantly different and not a direct restatement. ...
1
vote
1answer
55 views

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult?

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult? I am consider taking a undergraduate course in my college called mathematics of statistics and in the ...
0
votes
2answers
99 views

Metrics and the Kuratowski closure axioms

Edit: Succinct proofs from user87690 can be found below, but I will gladly up-vote other valid approaches to any of the problems here! The following questions concern closure operators and the ...
0
votes
2answers
51 views

Proving $P(A|(B \cap C)) = P(B | (A \cap C)) P(A | C) / P(B | C)$ using Bayes' theorem.

The following equation can be proven rather uglily, provided that $P(B \cap C)$, $P(A \cap C)$ and $P(C)$ are non-zero, by expanding the conditional probabilities. $$P(A | (B \cap C)) = \frac{P(B | ...
1
vote
2answers
80 views

Prove that if p divides xy then p divides x or p divides y

I am given that the following proposition is true. (Proved in class) "Suppose that $x$, $y\in \Bbb Z$, not both zero. Then there exists $m$, $n\in\Bbb Z$ such that $$mx + ny = d$$ where $d$ is the ...
1
vote
2answers
44 views

If $G$ is simple, then $\epsilon \leq {v \choose 2}$ - Bondy/Murty - Graph Theory with Applications Page 4

Question: Does this proof hold? Is this a bad proof? Any nicer proofs that don't rely on other theorems? Notation: $\epsilon$ - Number of edges $v$ - Number of vertices G - Here, any Graph ...
2
votes
0answers
132 views

What more can i do to this infinite sum?

This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following: $$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi ...
4
votes
3answers
61 views

If d is a norm on V, is $\frac{d(x,y)}{1+d(x,y)}$ a norm on V?

Let d be a norm on a vector space V and let $\psi:V \to [0,\infty)$ be a function defined as $\psi(v)=\frac{d(v)}{1+d(v)}$. Is $\psi$ a norm on $V$? It seems that $\psi$ does not satisfy the ...
0
votes
1answer
171 views

Set of open intervals in R with rational endpoints is a basis for standard topology on R

Show that the set $\mathcal{B} = \{(a,b) \subset \mathbb{R}: a,b \in \mathbb{Q}\}$ is a basis for the standard topology on $\mathbb{R}$ First I'll show that $\mathcal{B}$ is a basis on ...
6
votes
1answer
88 views

Is $\int f=f-1\iff f(\cdot)=e^{\cdot}$ proved this way correct?

I saw this on math overflow and made me wonder, why does it work, is it rigorous, can we really factor like this, and where can we use similar tricks; Let $\int$ denote $\int_0^x$ Then solve $$\int ...
6
votes
0answers
128 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
5
votes
1answer
39 views

Set of all subsets of X that contain a set Q is a topology

Let $X$ be a set such that $Q \subset X$. Show that $\tau = \{\emptyset\} \cup \{U \in \mathcal{P}(X): Q \subset U\}$ is a topology on X. $\emptyset \in \tau$ by definition and $X \in \tau$ ...
1
vote
0answers
42 views

Abelian Group (Alternative Proof)

Is there an alternative method to prove $(ab)^{2}=a^{2}b^{2}$ for all elements $a,b \in G \implies$ $(ab)^{-1}=a^{-1}b^{-1}$ for all elements $a,b \in G$. then the one I give below? Let $G$ be ...
1
vote
0answers
41 views

Proof Verification/Alternative to Induction- Well ordering proof

This question grew out of Induction and Maximum Principle, which yours truly asked Sep 23, at 11:51. Due to Mauro Allegranza's suggest, I changed focus so as to first prove the equivalence of $(a)$ ...
1
vote
2answers
42 views

Proofs of identity for product of binomial coefficients

While verifying my answer to another question, I came across a problem of binomial coefficients: Does $\hspace{.2cm}\displaystyle \prod_{k=1}^{n-1}\binom{n-1}{k}=\prod_{k=1}^{n-1}k^{2k-n}$ for all ...
0
votes
0answers
55 views

what is the probability that there is a string of k consecutive heads?

A coin is flipped n times. Assuming that the flips are independent, with each one coming up heads with probability p, what is the probability that there is a string of k consecutive heads? An answer ...
2
votes
4answers
119 views

Prove that a continuous function defined on an interval $[a,b]$ has a fixed point.

I have to prove that : Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$. But I don't know how to attack it since I can't apply anything of ...
1
vote
0answers
59 views

Exponential Function Limit Question

When I was first introduced to a derivation of the Taylor series representation of the exponential function here (pg 25): I noted the author, Dunham mentioning that the argument was non-rigorous. I ...
1
vote
1answer
38 views

Residue Calculus (Computing an Improper Integral)

Use residue calculus to compute the integral $\int_{-\infty}^{\infty}\frac{1}{(z^{2}+25)(z^{2}+16)}dz$ My solution If we add to the interval $I_{R}=[-R,R]$ add the semicircle $\gamma_{R}$ in the ...
2
votes
3answers
61 views

Applying Rouché's Theorem

Determine how many zeros of the following polynomial lie inside the circle $|z|=2$ \begin{equation} z^{5}+2z^{4}+z^{3}+20z^{2}+3z-1=0\end{equation} My Reasoning If we put $f(z)=z^{5}+2z^{4}$ and ...
0
votes
0answers
45 views

Proving that the Poisson random variable with mean $x$ is the Poisson random variable $X$ with maximum value of $\text{Pr}(X=x)$.

For any pair of positive numbers $\mu_1$ and $\mu_2$, let $X_1$ be the Poisson random variable with mean $\mu_1$, and $X_2$ be the Poisson random variable with mean $\mu_2$. Proof that $\text{Pr}(X_1 ...
0
votes
1answer
39 views

How to solve this *without* handshake theorem?

Suppose $45$ handshakes occurred in a room, how many people were in the room? Someone asked me this question and I was going to answer him using graph theory and my knowledge of the number of ...
-1
votes
2answers
27 views

Prove radius chord theorem without using congruent traingles

Suppose that $P(a,b)$ and $Q(c,d)$ are two points on the unit circle $x^2 + y^2 = 1$, and let $M$ be the midpoint of chord $PQ$. (Without using congruent triangles), prove that $OM$ is perpendicular ...
1
vote
4answers
126 views

Constructive proof for existence of integer part of real number

I try to prove de following exercise of my analysis textbook. Show that for every real number $x$ there is exactly one integer $N$ such that $N \le x < N + 1$. I have been finding a ...
1
vote
1answer
96 views

alternating sum of zeta functions minus one is one half

During my work on a different infinite series I happened to prove that $\displaystyle\sum_{k=2}^{\infty}(-1)^k (\zeta(k)-1)=\frac{1}{2}$ where $ ...
1
vote
1answer
259 views

Show that | and $\downarrow$ are the only binary connectives \$ such that {$} is functionally complete.

I've been reading and coping with van Dalen's Logic and Structure for a a few days. However, I've getting problems to solve his Exercise 6 from Ch 1 Sec 1.3 (p.28). In this exercise, van Daken asks ...
1
vote
1answer
76 views

Beppo-Levi: Reverse

For a merely decreasing positive sequence it fails: $$f_n:=\frac{1}{n}\chi_{[n,\infty)}:\quad\int f_n\mathrm{d}\lambda=\infty\nrightarrow0$$ For a dominated decreasing positive sequence it holds: ...
2
votes
2answers
78 views

Alternate proof for $a^2+b^2+c^2\le 9R^2$

As I studying geometric inequalities, one of those famous inequalities is $$a^2+b^2+c^2\le 9R^2$$ I did some research and I found that there is a proof (not exactly the this inequality but an useful ...
1
vote
0answers
23 views

Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
1
vote
0answers
26 views

Basic calculus question with continuous function [duplicate]

This is actually not my question, it was asked yesterday by user176744 in this link $[0,n]$ continuous function problem and I feel as if it didn't get enough attention. I am also interested in a ...
0
votes
1answer
61 views

Two reasons why $\int^{1}_{0}f(x) \,dx$ exists?

Consider $f$ on $[0,1]$ defined as $f(0)=0$ $$f(x)=2^{-n}\quad \text{if}\quad 2^{-n-1}<x\le2^{-n},$$ for $n=0,1,2,3,...$ I'm looking for two reasons why $\int^{1}_{0}f(x) \,dx$ exists? One ...
0
votes
1answer
47 views

how to prove the only difference between antidrivaties of a function is in their constants?

how to prove "If F is an antiderivative of f on an interval I , then the most general ...
10
votes
4answers
207 views

Inequality $\binom{2n}{n}\leq 4^n$

I would like to prove the following inequality, for $n=0,1,2,...$, $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
1
vote
2answers
49 views

Group of order $pq$ with $p\not\mid (q-1)$

Let $p, q$ be prime numbers, with $p<q$. If $G$ is a group of order $pq$ and $p\not\mid (q-1)$, then $G\cong \mathbb{Z}/pq\mathbb{Z}$. The standard way to prove this fact is using Sylow theorems, ...
4
votes
1answer
181 views

Will this algorithm stop before time?

For every $n \in \mathbb N$, let's define $a_0 = 0$, $$\begin{cases} a_{i+1} = 2a_i + 1 \pmod {2^n}, &\text{if it never appeared before} \\ a_{i+1} = 2a_i \pmod {2^n},& ...
0
votes
1answer
771 views

Proving graph connectedness given the minimum degree of all vertices

I know that this is a repeat of a previous question asked with a similar title, but I didn't want to revive an old thread. The solution presented in that thread seems to be the common one, but I was ...
0
votes
1answer
72 views

Prove that the graph $H = H_1\cup H_2 = (V_1\cup V_2,E_1\cup E_2)$ is connected.

Let $G = (V,E)$ be a graph and let $H_1 = (V_1,E_1)$ and $H_2 = (V_2,E_2)$ be two connected subgraphs of $G$ that have at least one node in common. Prove that the graph $H = H_1\cup H_2 = (V_1\cup ...