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

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2
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0answers
61 views

Alternative proof of de l'Hospital theorem

I see that the usual proof of de l'Hospital theorem involves the mean value theorem and is carried on in a case-by-case fashion. Could you point out some other proofs of de l'Hospital that are ...
10
votes
4answers
462 views

Construct quadrangle with given angles and perpendicular diagonals

The following came up when I worked on the answer for a different question (though it was ultimately not used in this form): Proposition. Given positive angles $\alpha,\beta,\gamma,\delta$ with ...
2
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3answers
324 views

Commutative artinian ring is noetherian

Suppose R is a commutative Artinian ring then R is Noetherian. I am aware of the proof which uses the idea of filtration. But I would like to prove this fact without that idea but haven't got far ...
2
votes
2answers
51 views

Disk/Washer method proof checking

This is a homework question, but i am just checking if what i am saying is correct, The question in the book states that A sphere of radius $r$ is cut by a plane of $h$ ($h < r$) units above the ...
2
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5answers
719 views

How to prove that there are infinitely many primes without using contradiction

How can I prove that there are infinitely many primes without using contradiction? I know the proof that is (not) by Euclid saying there are infinitely many primes. It assumes that there is a ...
1
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1answer
45 views

Identity Tranformation Proof- Is this enough to prove this statement?

Let {v$_1$,...,v$_n$} be a basis for a vector space V and let T:V$\to$V be a linear transformation. Prove that if T(v$_1$)= v$_1$,...,T(v$_n$)= v$_n$, then T is the identity tranformation on V. I'm ...
1
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5answers
66 views

Concise induction step for proving $\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$

I recently got a book on number theory and am working through some of the basic proofs. I was able to prove that $$\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$$ with the help of the identity ...
2
votes
1answer
100 views

A matrix-free way to find a fan basis of $V$?

Let $f:V\to V$ be a linear map, $\dim V =n$. A basis $( v_1, \ldots, v_n)$ of $V$ such that for all $j=1, \ldots,n$ the space $\text{span}(v_1,\ldots,v_j)$ is $f$-invariant is called a fan basis of ...
1
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0answers
29 views

Gradient points in the direction of greatest change

Can anyone provide me with an alternative, possibly more intuitive proof of this proposition? I'm confused with where $cos\theta$ has come from?
3
votes
1answer
102 views

Proof Verification of Schröder–Bernstein theorem

So I've spent some time studying the Schröder–Bernstein theorem, but I'm trying to do the exercise in "Naive Set Theory" by Paul Halmos regarding the theorem. The exercise is finding an alternative ...
1
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3answers
103 views

Evaluate $\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$ [duplicate]

Evaluate $$ \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$$ where $n\in\mathbb{N}$ This one is another intriguing question from my worksheet. I'm only allowed to use ...
3
votes
2answers
66 views

Prove every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.)

Let $A$ be a commutative ring with unity. Prove: Proof every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.) In the question before ...
1
vote
1answer
98 views

Cutting chocolate diagonally

Given is chocolate with rectangular pieces of size $a \times b$. If cut diagonally, how many pieces will it be split into? If knife passes exactly by co-catenating we assume there is no damage to ...
1
vote
1answer
87 views

Application of Contraction Principle?

Show that there is a unique solution to the equation \begin{equation} \frac{df}{dx}=(f(x)+x)x \tag{*}\end{equation} for $0 \le x \le 1$ and $f(0)=0$. Clearly, $(*)$ is a first order linear ...
0
votes
2answers
129 views

Proof of Lindelöf Theorem

I have been surfing the net to read the proof of the Lindelöf Theorem: Let $U\in \mathbb{R}^n$ be open and $U=\bigcup_{\lambda \in \Lambda} U_{\lambda}$where $\Lambda$ is an index set, ...
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2answers
54 views

Proof of Bézout's identity - Cohn - CA p26

Given two integers $a$ and $b$, there exist integers $u$ and $v$ such that $$au+bv=1$$ if and only if $a$ and $b$ are coprime. Attempt Proof: Assume $a$ and $b$ are not coprime, e.g. $a=kb,k\in ...
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0answers
54 views

Show that this Hypergeometric Function equal to this gamma function

I have a question related to hypergeometric functions: Show that ...
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2answers
70 views

Suppose G is a group, p is prime , Then the number of elements of G of order p is multiple of (p-1) [closed]

I need Help . "Suppose $G$ is a group, $p$ is prime , Then the number of elements of G of order $p$ is multiple of $(p-1) $". Give me any advise or note
3
votes
0answers
48 views

Topological proof for this set theory statement

Let $\mathcal{A}$ be an algebra of set (in a space $X$), such that any subcollection of disjoint sets in $\mathcal{A}$ is finite. Prove that $\mathcal{A}$ is finite. I already found a boring brute ...
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2answers
51 views

If H and K are finite subgroups of G (another proof )

I have a question and it's solution , but I want another proof if there exist . Thanks
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1answer
47 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
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votes
2answers
62 views

$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$ for polynomials

Prove that there exists constant $C>0$ that for all $f \in P_n$ we have: $$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$$ Where $P_n$ is space of polynomials with ...
0
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0answers
27 views

Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
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3answers
109 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
1
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1answer
33 views

Looking for a way to improve my inductive proof of a statement derived by Rolle's Theorem

The following problem is 'absolutely' clear: Problem: Let $f$ be continuous on the interval $[a,b]$ and $n$-times differentiable on $(a,b)$ and $f$ vanishes on $n+1$ points $x_0< x_1 < \dots ...
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1answer
73 views

Accounts of the proof of Fermat's Last Theorem

I would like to collect a set of references to pieces of Wiles' 1995 proof of Fermat's Last Theorem. Has anyone recompiled the proof into another paper? Are there any books or articles that describe ...
5
votes
2answers
387 views

A Deviation from a Conventional Proof of the Basel Problem

There's been many topics on the Riemann-Zeta function, specifically $\zeta(2)$.$$\zeta(2)=\sum_{n=1}^\infty\frac{1}{n^2}=\int_0^1\int_0^1\frac{1}{1-xy}dA$$This is the Basel Problem. Taking the ...
4
votes
3answers
100 views

Prove $-1$ and $1$ are the only units in $\mathbb{Z}$

Prove $\mathbb Z^*=\{-1,1\}.$ I have a proof, which is posted as an answer below. I'm looking for an alternate proof.
2
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1answer
94 views

An alternative proof of the Tietze Extension Theorem(s)

Last summer I was working through a lot of Topology. I made it through the sections of my notes that dealt with separation properties, covering properties and continuous functions between spaces ...
3
votes
2answers
39 views

What is $\gcd(x,x+2)$?

Show that $\gcd(x,x+2)$ is $1$ if $x$ is odd and $2$ if $x$ is even. I am looking for a much simpler proof beside the one which I have posted.
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0answers
36 views

Verify why a thing of if this proof is right

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some ...
0
votes
0answers
39 views

Essential part to undestand a proof . [duplicate]

In the proof of the the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
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1answer
36 views

Question about a proof concerning abelian p-groups

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for ...
2
votes
2answers
81 views

A doubt with a part of a certain proof.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
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1answer
39 views

Why do a coset is isomorphic to a certain set.

I have encountered with the proof of the next lemma suppose G is a finite abelian p-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
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2answers
127 views

An alternative proof for sum of alternating series evaluates to $\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$

How does one prove the given series? $$\sum_{n=0}^\infty\left(\frac{(-1)^n}{4n-a+2}+\frac{(-1)^n}{4n+a+2}\right)=\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$$ This series came up in xpaul's ...
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1answer
172 views

Proof of easy matching condition for Hall's theorem

I was studying with the recitations provided in the course 6.042 "Mathematics for Computer Science" of MIT OCW and while studying the proof of Hall's marriage problem, I understood the first proof ...
5
votes
1answer
201 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 ...
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9answers
379 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
23 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 $u$ ...
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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?
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1answer
110 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
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1answer
93 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 ...
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votes
12answers
745 views

What are the theorems in mathematics which can be proved using completely different ideas?

I would like to know about theorems which can give different proofs using completely different techniques. Motivation: When I read from the book Proof from the Book, I saw there were many ...
1
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0answers
32 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
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1answer
77 views

Special Integral Proof

How to prove $$\int_0^\infty x^{2n-1} \exp(-a^{x^3})\, dx = \frac{\Gamma(n)}{2a^n} ,\quad n> 0 ,\quad a>0. $$
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1answer
46 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 ...
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votes
1answer
44 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
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1answer
46 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
53 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 ...