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

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31 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 ...
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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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2answers
50 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 ...
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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 ...
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6answers
141 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 ...
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24 views

Geometric interpretation or solution of an induction problem

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, ...
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2answers
40 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 ...
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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 ...
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59 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
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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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 ...
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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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4answers
375 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 ...
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0answers
96 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 ...
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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 ...
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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 ...
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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 ...
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1answer
72 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} ...
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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
43 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 ...
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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 ...
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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. ...
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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 $f(x) \geq 0\quad\forall x \in \mathbb{R}$. ...
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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 ...
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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
64 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 ...
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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 $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ ...
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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 ...
2
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1answer
21 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 $PQ \parallel AD$. ...
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1answer
387 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 ...
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0answers
55 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 ...
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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 ...
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3answers
70 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 ...
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2answers
27 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 ...
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5answers
639 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 ...
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1answer
18 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 ...
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3answers
33 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 ...
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1answer
57 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 ...
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0answers
23 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?
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1answer
57 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 ...
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3answers
92 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 ...
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2answers
40 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 ...
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1answer
77 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 ...
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1answer
44 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 ...
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2answers
30 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, ...
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2answers
44 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
44 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
57 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
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0answers
42 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
47 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