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

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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, ...
2
votes
1answer
37 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, ...
7
votes
5answers
898 views

Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ...
0
votes
0answers
16 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_{1\leqslant j \leqslant ...
3
votes
1answer
17 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
45 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
61 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 ...
1
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2answers
56 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 ...
0
votes
1answer
56 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
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0answers
46 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 ...
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0answers
27 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 ...
3
votes
6answers
144 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 ...
16
votes
2answers
1k views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as briefly as possible

Can someone provide the proof of the special case of Fermat's Last Theorem for $n=3$, i.e., that $$ x^3 + y^3 = z^3, $$ has no positive integer solutions, as briefly as possible? I have seen some ...
2
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1answer
44 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
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1answer
63 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 ...
21
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1answer
495 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: ...
12
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0answers
98 views

$\log_2 13$ is irrational.

$\log_2 13$ is irrational. Is it true? $x=\log_2 13$ $\implies 2^x=13$ 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, $13$ is ...
0
votes
2answers
42 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 ...
0
votes
1answer
365 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 ...
1
vote
1answer
44 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
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1answer
17 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 ...
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0answers
60 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?
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1answer
25 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 ...
10
votes
4answers
377 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
9answers
786 views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
1
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1answer
25 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
26 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 ...
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votes
3answers
3k views

Sum of irrational numbers

Well, in this question it is said that $\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$, and the owner asks for "alternative proofs" which do not use rational root theorem. I wrote an ...
2
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2answers
30 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
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0answers
27 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 ...
4
votes
1answer
73 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} ...
5
votes
2answers
138 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 ...
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1answer
337 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 ...
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2answers
67 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 ...
16
votes
1answer
388 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 ...
12
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1answer
1k views

Direct aproach to the Closed Graph Theorem

In the context of Banach spaces, the Closed Graph Theorem and the Open Mapping Theorem are equivalent. It seems that usually one proves the Open Mapping Theorem using the Baire Category Theorem, and ...
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0answers
26 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
82 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
1answer
256 views

Is there some elementary proof of invariance of domain?

Invariance of domain at least in statement seems a simple result. I mean, the first time I saw the statement I thought: "the proof can't be that bad", but when I searched for it I saw that it needs ...
2
votes
1answer
163 views

Poincaré-Hopf theorem using Stokes

The wiki entry on the Poincaré-Hopf theorem claims that it "relies heavily on integral, and, in particular, Stokes' theorem". However, in the sketch of proof given there which is more or less the one ...
7
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0answers
55 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 ...
3
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0answers
171 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
4
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1answer
177 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},& ...
1
vote
3answers
62 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 ...
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2answers
66 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 ...
1
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1answer
159 views

Showing a topology is not metrizable

Show $\prod_{N} \mathbb{R}$ with the box topology is not metrizable. The Box Topology on $\prod_{j \in J} X_j$ ($X_j$ topological spaces) is generated by the basis $\left\{\prod_{j \in J} U_j \; ...
5
votes
3answers
90 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 $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ ...
4
votes
1answer
69 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 ...
8
votes
4answers
409 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
8answers
421 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 ...