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

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8
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1answer
136 views

“Novel” proofs of “old” calculus theorems

Every once in a while some mathematicians publish (mostly on the Montly) a new proof of an old (nowadays considered "basic") result in analysis (calculus). This article is an example. I would like ...
8
votes
4answers
377 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 ...
1
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8answers
387 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
2
votes
3answers
1k views

Prove that in a parabola the tangent at one end of a focal chord is parallel to the normal at the other end.

Prove that in a parabola the tangent at one end of a focal chord is parallel to the normal at the other end. Now, I know prove this algebraically, and that's very easy, but I am not getting any ...
2
votes
1answer
20 views

Point in a rectangle

$ABCD$ is a rectangle and $P$ is a point in the same plane. If the perpendicular through $C$ to $AP$ and the perpendicular through $B$ to $DP$ intersect at $Q$, prove that $PQ \parallel AD$. ...
4
votes
1answer
164 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},& ...
2
votes
0answers
51 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 ...
1
vote
1answer
71 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: ...
1
vote
1answer
75 views

Measure of Elementary Sets Proof

I am struggling with what seems like a very simple problem from Terrence Tao's Introduction to Measure Theory book (which is available for free online by the way). What I am trying to prove is the ...
1
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6answers
253 views

Solution of $\dfrac{a}{b}=\dfrac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$

Let $\dfrac{a}{b}=\dfrac{a'}{b'}$ , $a,b,a',b' \in \mathbb{N}$ s.t. $a$ and $b$ have no common factors. How can we show that the only solution to this equality is $a'=na$ and $b'=nb$, $n$ is a natural ...
1
vote
3answers
57 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
5answers
627 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 ...
2
votes
2answers
21 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 ...
1
vote
1answer
309 views

Problem with alternate solution — Equation of plane through point and containing intersection line of two planes [Stewart P $803, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
5
votes
2answers
270 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 ...
1
vote
1answer
51 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
vote
1answer
16 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 ...
0
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3answers
31 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 ...
1
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0answers
21 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
49 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
vote
3answers
84 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 ...
1
vote
1answer
74 views

Cutting chocolate diagonally

Given is chocolate with rectangular pieces of size $a \times b$. If it will be cut diagonally, how many pieces will be splitted? If knife pass exactly by concatenating we assume there is no damage ...
1
vote
1answer
89 views

Find the coefficients such that all four roots of $(x^2-px+q)(x^2-qx+p)$ are natural numbers

Find all ordered pairs $(p,q)$ of natural numbers such that all $4$ the roots of $$f(x)=(x^2-px+q)(x^2-qx+p)$$ are natural numbers. I got a solution of the problem (see below) but I want some ...
2
votes
2answers
37 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
41 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 ...
1
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2answers
43 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 ...
0
votes
2answers
22 views

Proof of Lindelof Theorem

I have been surfing the net to read the proof of the Lindelof Theorem: Let $U\in \mathbb{R}^n$ be open and $U=\bigcup_{\lambda \in \Lambda} U_{\lambda}$where $\Lambda$ is an index set, ...
1
vote
2answers
54 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
0
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2answers
44 views
0
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1answer
40 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 ⇒ ...
3
votes
0answers
39 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 ...
6
votes
2answers
49 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 ...
7
votes
1answer
369 views

Proof that a certain entire function is a polynomial

Let $n\in\mathbf{N}$ be fixed, and $f$ entire and $|f^{-1}(\left\lbrace w\right\rbrace)|\leq n$ for every $w\in\mathbf{C}$. Then $f$ is a polynomial of degree at most $n$. I try to prove this ...
0
votes
0answers
11 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' ...
7
votes
3answers
83 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 ...
16
votes
9answers
352 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 ...
1
vote
1answer
23 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 ...
20
votes
1answer
473 views

Verify matrix identity $A^tD-C^tB=I$ on certain hypotheses

Given $n\times n$ real matrices $A,B,C,D$ such that: $AB^T$ and $CD^T$ are symmetric $AD^T-BC^T=I$ Prove that $A^TD-C^TB=I$ The solution I have come up with after a very long time is to consider: ...
1
vote
1answer
48 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
1answer
136 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 ...
1
vote
3answers
64 views

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

Prove $\mathbb Z^*=\{-1,1\}.$ I have a proof, which is posted as an answer below. I'm looking for an alternate proof.
1
vote
1answer
44 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
34 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.
0
votes
1answer
33 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 ...
1
vote
0answers
35 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 ...
2
votes
2answers
78 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
votes
0answers
38 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 ...
3
votes
3answers
86 views

Could this linear algebra proof be done without computation?

From page 95 of Hoffman & Kunze's Linear algebra: Let $T$ be the linear operator on $\mathbb{R}^2$ defined by $T(x_1,x_2)=(-x_2,x_1)$ Prove that if $B$ is any ordered basis ...
0
votes
1answer
37 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 ...