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
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1answer
34 views

Measure theoretic proof of $|\Bbb{Z}^d/A\Bbb{Z}^d| = |\det(A)|$

Let $A \in \Bbb{Z}^{d\times d}$ be an invertible matrix with entries in $\Bbb{Z}$. It is well-known (and can be proved using algebraic properties of matrices) that the index of the group $A \Bbb{Z}^d ...
1
vote
0answers
31 views

Prove without method of contradiction that there exists a real number less than every positive real number that is positive

This question was asked before for proof by contradiction and which got me into thinking whether i could prove it without using a contradiction Original problem statement is here Prove by ...
1
vote
1answer
44 views

Proving continuity of $f(x)=x\cos(2\pi/x)$ at $x=0$

I know that the function $f(x)=x\cos(2\pi/x)$ if $x\neq0$ and $f(0)=0$ is continuous at $x=0$ using $\epsilon-\delta$ as follows: $\lvert x\cos(2\pi/x)\rvert=\lvert ...
2
votes
1answer
53 views

An inequality with elementary symmetric polynomials

Fix a natural number $n\geq 1$. Let $a_1, \ldots, a_n$ be $n$ real numbers such that $a_i>0$ for each $i$. Show that for each natural $k$ with $0\leq k\leq n$ $$e_k(a_1,\ldots, ...
1
vote
0answers
141 views

Partition of unity of Lagrange polynomials

Given a sequence of increasing real numbers $T = \{t_1 < t_2 < ... < t_{d+1}\}$, the $d+1$ Lagrange polynomials $L_i(t)$ of degree $d$ are defined as $$L_i(t) = \prod_{\substack{1\leqslant j ...
1
vote
1answer
28 views

Other proof for existence of monotone subsequences

Is there any other proof of Bolzano-Weierstrass theorem (i.e.: Let ${\{x_n}\}$ be an arbitrary sequence of real numbers. Then ${\{x_n}\}$ has a monotone subsequence.), WITHOUT using concept of ...
1
vote
2answers
68 views

Alternative proof of a transpose property

I am asked to prove; $$(AB)^T=B^TA^T$$ although it is very simple to prove it by the straight forward way, in the exercise I am asked to prove it without using subscripts and sums, directly from the ...
2
votes
2answers
73 views

Is there another way to prove $(x-n)^2 = (n-x)^2$

Let's say $n$ is $4$. So, I came up with the solution below. $(x-4)^2 = (x-4)(x-4) = x^2 - 8x + 16$ $(4-x)^2 = (4-x)(4-x) = 16 - 8x + x^2 = x^2 - 8x + 16$ I was wondering if there is another way ...
0
votes
1answer
91 views

Proving Submultiplicativity on a Matrix Norm

Let $||A||=(\sum_{i=1}^{n}\sum_{j=1}^{n}{a_{ij}^p})^{1/p}$, and let p=2. Then prove that $\|AB\|\le \|A\|\|B\|$ I have looked at numerous proofs for this, and I don't see one that satisfies me ...
2
votes
0answers
52 views

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices? [duplicate]

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices or having to choose a particular base ? Such a proof should probably use a non matricial definition of traces. One ...
1
vote
2answers
63 views

How to get to $5^3 \geq n^3$ in the proof by contradiction?

This is the same problem asked here. - Next step to take to reach the contradiction? Here is it again. I understand the solution - how you want to get to the fact 100 divides n^2 and then go ...
1
vote
1answer
39 views

e Online source for alternative proofs

I'm looking for some alternative proofs for various theorems. My goal is to compile a list of various proofs each relating to a specific theorem (such as the triangle inequality, Fermat's Little ...
4
votes
6answers
193 views

Prove that $x-1$ is a factor of $x^n-1$

Prove that $x-1$ is a factor of $x^n-1$. My problem: I already proved it by factor theorem† and by simply dividing them. I need another approach to prove it. Is there any other third ...
12
votes
4answers
222 views

Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
14
votes
6answers
591 views

Fascinating induction problem with numerous interpretations

Problem: Suppose you begin with a pile of $n$ stones and split this pile into $n$ piles of one stone each by successively splitting a pile of stones into two smaller piles. Each time you split a pile, ...
0
votes
2answers
48 views

Trouble solving this induction problem

Show that, for every $n\ge2$, $3^n >n(n-1)$. Well, I started by showing the base case ($n = 2$): $3^2 > 2$ Now, for $n+1$: $P(n)\Rightarrow P(n+1)$ $$3^{n+1} > (n+1)n$$ My ...
1
vote
1answer
57 views

Can someone verify my proof by contraposition?

This is a problem from Discrete Mathematics and its Applications Is there a way to tell right away what type of proof to use or does that just come with practice (build intuition - oh here i ...
1
vote
0answers
173 views

New proofs of the Fundamental Theorem of Calculus

Apart from the standard one, are there any other proofs of the Fundamental Theorem of Calculus which have been published recently?
1
vote
1answer
21 views

An alternative proof to: $P_{[X,Z]}=P_X+M_XZ(Z'M_XZ)^{-1}Z'M_X$

With $P_A=A(A'A)^{-1}A'$ and $M_A=I-P_A$ denoting the usual orthogonal projection matrices, I'm trying to find an alternative proof to $$ P_{[X,Z]}=P_X+M_XZ(Z'M_XZ)^{-1}Z'M_X.\tag{i} $$ I already ...
1
vote
1answer
36 views

Next step to take in direct proof for one to one?

This is from Discrete Mathematics and its Applications And the definition of strictly increasing. Here is my work so far. I know that a direct proof involves making an assumption p, which in ...
2
votes
2answers
50 views

Help with a Royden exercise of measure

I'm solving the exercise 12, of section 4 The General Lebesgue Integral from the Royden's book Real Analysis 3rd edition: Let $g$ be an integrable function on a set $E$ and suppose that $(f_n)$ is a ...
11
votes
4answers
592 views

How can I complete this proof by contradiction?

This problem is from Discrete Mathematics and its Applications: Prove that there are no solutions in integers $x$ and $y$ to the equation $2x^2 + 5y^2 = 14$. I am trying to use proof by ...
21
votes
1answer
224 views

$\log_2 13$ is irrational

Is it true that $\log_2 13$ is irrational? Let $x=\log_2 13\implies 2^x=13$. So, it will be an irrational number, if not,$$x=\frac p q$$ and $$2^{\frac p q}=13$$ $$\implies 2^p=13^{q}$$ Since, ...
2
votes
2answers
42 views

Next step to take to reach the contradiction?

This problem is from Discrete Math and its Applications I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book Here is my work so far for ...
1
vote
0answers
42 views

Prove that the maximum in absolute value of any monic real polynomial of n-th degree on [-1, 1] is not less than $\frac{1}{2^{n-1}}$

One solution is: Note that equality holds for a multiple of the n-th Chebyshev polynomial $T_{n}(X)$ The leading coefficient of $T_{n}$ equals $2^{n-1}$, so $C_{n}(X) = \frac{1}{2^{n-1}}T_{n}(X)$ is ...
0
votes
1answer
30 views

How to introduce bi-conditional in this proof?

This is from Discrete Mathematics and its Applications Just for context, I know that the universal set is everything and that the complement of A is just difference of the universal set and A. A ...
4
votes
1answer
89 views

Wedge product of a direct sum and the Yoneda Lemma

In a comment to http://math.stackexchange.com/a/344851/58601, Martin Brandenburg suggests that one may prove the existence of the canonical isomorphism $\wedge^n(W_1 \oplus W_2) \to \bigoplus_{p+q=n} ...
6
votes
2answers
144 views

Difficulties understanding a proof of $\int_0^{\infty} \frac{\sin(x)}{x} \, dx = \frac{\pi}{2}$

I got a homework and I've trying to do this problem about 2 days, but I "lost my fight". So I turn to you. I have to prove that $$\int _0^\infty \frac{\sin (x)}{x} \, dx = \frac{\pi}{2}.$$ I can't use ...
2
votes
1answer
57 views

An integer square matrix of prime order has size at least $(p-1)\times (p-1)$

There's$\let\geq\geqslant\DeclareMathOperator{\GL}{GL}$ this exercise in my algebra course book: Let $p$ be a prime and $A\neq I$ an $n\times n$ matrix over $\mathbb Z$ such that $A^p=I$. Prove ...
1
vote
2answers
69 views

How can I prove this using number theory only

So this book I'm reading has this question: show that if $(a,n)=(b,n)=1$ the the equation $$ax+by\equiv c(mod( n))$$ has exactly $n$ different solutions. I was only able to prove it using ...
0
votes
0answers
27 views

Is the proof rigorous enough?

Proposition Let $n$ be a Natural Number and let $P(n)$ be a property pertaining to the Natural Numbers such that whenever $P(m{++})$ is true, $P(m)$ is true. Suppose that $P(n)$ is true. ...
2
votes
4answers
88 views

Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$

Show that $$\displaystyle1 - \frac{x^2}2\leq\cos x\quad\forall x \in \mathbb{R}$$ Let $f(x) = \cos x - 1 + \frac{x^2}2$; then we need to show that $f(x) \geq 0\quad\forall x \in \mathbb{R}$. ...
7
votes
0answers
71 views

Largest rectangle bounded under a function

Let $f$ be a positive monotonically increasing real function in $[0,1]$. Let $F$ be the area under the curve of $f$ ($F=\int_0^1{f(x)dx}$) For every $x\in[0,1]$, let $G(x)=f(x)*(1-x)$ = the area of a ...
1
vote
3answers
67 views

Check proof that operator in unbounded please

I have to show that $f:\mathcal{C}'[a,b]\rightarrow \mathbb{R}$ with $f(x)=x'(\frac{a+b}{2})$ is unbounded. Here $\mathcal{C}'[a,b]$ (the space of continuously differentiable functions) is to be ...
1
vote
2answers
92 views

Show that $[2x]+[2y] \geq [x]+[y]+[x+y]$

Prove that $[2x]+[2y] \geq [x]+[y]+[x+y]$ whenever $x$ and $y$ are real numbers. The $[]$ symbol is the greatest integer or floor function. I have proved this fact by cases, but I stumbled upon what ...
6
votes
3answers
95 views

Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.

I can show that for $x > 0$ and $r_{i} > 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ ...
5
votes
1answer
87 views

A combinatorial proof of Wilson's Theorem

I am looking for a combinatorial proof of Wilson's Theorem. Something along the lines of this kind of proof. $\textbf{Combinatorial proof of Fermat's Little Theorem}$ First consider a $p$ -tuple and ...
2
votes
1answer
25 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 $PQ \parallel AD$. ...
18
votes
1answer
427 views

“Novel” proofs of “old” calculus theorems

Every once in a while some mathematicians publish (mostly on the American Mathematical Monthly) a new proof of an old (nowadays considered "basic") result in analysis (calculus). This article is an ...
2
votes
0answers
60 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
453 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
votes
3answers
226 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
46 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
votes
5answers
705 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
vote
1answer
31 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
vote
3answers
44 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
97 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
0answers
28 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
88 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
99 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 ...