Questions tagged [alternative-proof]
If you already have a proof for some result but want to ask for a different proof (using different methods).
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Is there an alternative way to solve this trivial problem?
I dealing with a trivial problem:
Let $f(x) = xy$ be some area. Given $1500 = 15x + 6y$, find $x$ and $y$ so that the area is maximized.
Usually pretty easy when you can just plug in $(1500 - 15x)/6$...
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Is this proof circular or is it valid?
I am playing with the chernoff bounds and derived a bound from its additive version. The method I used seems a little suspicious and circular in logic, but I cant figure out what the mistake is.
...
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Proving the variational method for a nonhomogenous linear system of ODEs
I've been trying to prove the following as titled:
If I have two linearly independent independent solutions of a homogenous linear system then,
$x(t) = v_1(t)x_1(t) + v_2(t)x_2(t)$
$y(t) = v_1(t)y_1(t)...
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Are there nonzero natural numbers such that $\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}$?
Check if there are nonzero natural numbers $n,x,y$ such that:
$$\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}. $$Thank you in advance!
My ideas
So we can simply show that $4n+5,5n+1,9n+4$ are ...
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Proving the relation between sides and diagonals of parallelogram without trigonometry and Pythagoras theorem
I am looking for a way to prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides. However, I would like to know whether it is possible to ...
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Convergence of $\sum\limits_{k=1}^{\infty}\left(a^\frac{1}{k^2}-1\right)$ and $\sum\limits_{k=1}^{\infty}\left(a^\frac{1}{k}-1\right)$ for $a>1$
I am reading a real analysis book which in the chapter about sequences and series asks to prove that, for $a>1$, $\sum\limits_{k=1}^{\infty}\left(a^\frac{1}{k^2}-1\right)$ is convergent and $\sum\...
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$(\operatorname{Mat}_n R)[x] \cong \operatorname{Mat}_n R[x]$ in the language of Categories
I am currently working through Hungerford’s Algebra book. Question 2 on page 156 states:
Let $\operatorname{Mat}_n R$ be the ring of $n \times n$ matrices over a ring $R$. Then for each $n \geq 1$.
$$...
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If $f'(x)$ is continuous at $a$ then prove that $\lim_{h\to 0}\frac{f(a+h)-f(a-h)}{2h}=f'(a)$
Prove that, $\lim_{h\to 0}\frac{f(a+h)-f(a-h)}{2h}=f'(a)$ if $f'(x)$ is continuous at $a.$
This was how the question was presented in a regional book focusing upon Mean Value Theorems and no other ...
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Angle chasing problem with quadrilateral - possible approaches?
CONTEXT
I was recently working on some Langley-style problems, and wanted to construct some others, based on the "reverse engineering" approach I developed to solve them.
PROBLEM
The ...
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Alternative to disproving $\sum_{k=1}^{n} \frac{1}{k} < 3$ by exhaustion
I am faced with the following problem:
Consider the statement $\sum_{k=1}^{n} \frac{1}{k} < 3$. How to disprove this claim, for some $n$?
I have taken to disproving this by using a simple counter-...
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Prove $\lim\limits_{n\to+\infty}\left(\frac{n^2-1}{n(1+n^2)}\right)^\frac{1}{\sqrt{n}}=1$ without $\lim_{x\to 0^+}x\ln(x)=0$ and continuity of $e^x$
I am reading a real analysis book which in the chapter about sequences and series asks to prove that $\lim\limits_{n\to+\infty}\left(\frac{n^2-1}{n(1+n^2)}\right)^\frac{1}{\sqrt{n}}=1.$
I have proved ...
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In a group of order 120 with a normal subgroup of order 5, every subgroup of order 15 contains that normal subgroup. Simplest proof?
Let $G$ be a group of order $120$ with a normal subgroup $N$ of order 5. Let $H$ be any subgroup of $G$ of order $15$. Prove $N$ is a subgroup of $H$.
I have a proof (see below), but I am wondering: ...
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Alternative proof request that $Aut(\mathbb{Q}_p)=\{Id\}$
I am working through Neukrich's Algebraic number theory book, and just read the chapter introducing the p-adics. At this stage we know the definition of the p-adics as the fraction field of the ...
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Exercise 14, Section 2.4 of Hungerford’s Abstract Algebra
Corollary 4.9. If $H$ is a subgroup of index $n$ in a group $G$ and no non trivial normal subgroup of $G$ is contained in $H$, then $G$ is isomorphic to a subgroup of $S_n$.
Corollary 4.10. If $H$ is ...
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Proving statements vs disproving statements
This exercise came from my discrete mathematics book.
Determine which statements are true and which are false. Prove those that are true and disprove those that are false.
problem 6: $\frac{\sqrt 2}...
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Evaluate $\,\lim\limits_{x \to 0^{-}} \frac{\sin(\lfloor x\rfloor)}{x}$
How can I evaluate this limit.
$$\lim_{x \to 0^{-}} \frac{\sin(\lfloor x\rfloor)}{x}$$
This is what I did:
$x\in \big(\frac{-1}{2},0\big)\;$ ; $\;\;\lfloor x\rfloor =-1 $
$$\lim_{x \to 0^{-}}\sin(\...
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Probability Proof(Finan #5.1.15)
Show that $P(A|B)>P(A)$ if and only if $P(A^c |B)<P(A^c)$. We assume
that $0<P(A)<1$ and $0<P(B)<1$ ( Finan #5.1.5)
Taking the complement of probabilities reverses the sign of the ...
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Alternative Proof that ln(2) is Irrational
Is the following a correct proof that $\ln(2)$ is irrational without using the fact that $e$ is transcendental?
The power series $\ln(1+x)=\sum_{k=1}^\infty \frac{(-1)^{k+1}x^k}{k}$ for $x=1$ gives ...
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Simpler proof of identity $ \sum_{n=1}^{\infty} \frac{H_n^2}{n2^n} = \frac{7}{8} \zeta(3) $
The identity in question can be obtained by first proving
$$ \sum_{n=1}^{\infty} \frac{H_n^2}{n} z^n = - \frac{1}{3} \log^3(1-z) -\log(1-z) \text{Li}_2(z) + \text{Li}_3(z), \hspace{0.5cm} |z|<1, $$
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A simpler approach to Brezis' Proposition 8.5
I'm reading a result from Brezis' Functional Analysis
Proposition 8.5. Let $u \in L^p(\mathbb{R})$ with $1<p<\infty$. The following properties are equivalent:
(i) $u \in W^{1, p}(\mathbb{R})$,
...
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Evaluating $\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{ij(i+j)^2}$
I was playing around with double sums and encountered this problem: Evaluate
$$\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{ij(i+j)^2}$$
It looks so simple I thought it must have been seen before, ...
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Is every finite scheme affine?
$\def\sF{\mathcal{F}}
\def\sO{\mathcal{O}}$Today I wondered is every finite scheme affine? and I found a proof using Serre's criterion on affiness; however, I don't know if I may be using a ...
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Find the Length of Parallel Line Through Trapezoid's Diagonals Intersection with given Bases Lengths
The problem:
A straight line is drawn through the point of intersection of the diagonals of a trapezoid, which is parallel to the bases and intersects the sides of the trapezoid at points $M$ and $K$....
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Two different expressions from solving $s(n, m) = s(n-1, m) + s(n-1, m-1)$, alternate proof they are equal.
Before reading this please quickly look at this question I asked before, and the accepted answer. Upon further inspection, I found another solution to the recurrence relation, namely (the reciprocal ...
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A question about general associative law for intersection and union.
If $\mathfrak F$ is a partition of a collection $\mathcal A$ then it is not hard to show that the equality
$$
\tag{1}\label{eq: general associative law for intersection}\bigcap_{A\in\mathcal A}A=\...
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Alternative Proof for Surjections over finite groups
Let $G$ be a finite group & defined a one-one map $T$ in $G$ such that $$aT=a^{-1}\cdot aJ$$ where $J$ is an automorphism of $G$ with a property $aJ=a$ only if $a=e$.
I can prove that $T$ is well-...
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Is possibile to prove the equality $\left(⋃_{X∈\mathfrak X}X\right)×\left(⋃_{Y\in\mathfrak Y}Y\right)=⋃_{(X,Y)∈\mathfrak X×Y}(X×Y)$ using projections?
So I know that if $A$ is a subset of a set $X$ and if $X$ is a subset of a set $Y$ then the equalities
$$
\begin{equation}
\tag{1}\label{1}A\times B=\pi_X^{-1}[A]\cap\pi_Y^{-1}[B]
\end{equation}
$$...
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Finding an upward-opening catenary passing through $(x_1,y_1),(x_2,y_2)$ with length $L$
I want an equation $y=a\cosh(\frac{x-b}{a})+c$ for a catenary opening upward that passes through $(x_1,y_1)$ and $(x_2,y_2)$. WLOG, assume $x_2>x_1$.
Let $L$ be the length of the catenary between $(...
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Any other methods for evaluating $\int_{0}^{\frac{\pi}{4}}\frac{\arctan(\cos(x))}{\cos(x)}\,dx$
I evaluated the following integral:
$$I:=\int_{0}^{\frac{\pi}{4}}\frac{\arctan(\cos(x))}{\cos(x)}\,dx$$
I would like to see any alternate solutions, here is my work.
Using:
$$\frac{\arctan(x)}{x}\...
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The fourth power of any integer has the form $8m$ or 8m + 1 for some integer m.
In my textbook I came across this problem:
Prove that the fourth power of any integer has the form $8m$ or $8m + 1$ for some integer $m$.
I know how to solve the algebra but don't understand the ...
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To check the sign and convexity of a transcendental function using naive approach
I was solving this BVP:
$$y''+y= \csc(x), y(0)=0=y(\pi/2), x\in[0,\pi/2]$$
By following the same lines of this solution using the variation of parameters method, I got the solution:
$$y=-x\cos(x) + \...
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Is it possible to prove all main integral properties for the discreet sum integral formula?
There are a ton of different formulas for integrals, and my favourite is one I often hear called the discreet sum integral, defined such that
$$\int_a^bf(x)dx=\lim_{d\to0^+}\sum_{x=a/d}^{b/d}df(dx)$$
...
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An alternative definition of the limit of a function
In a course on real analysis one usually comes across the definition of the limit of a function:
Given a function $f:A\to \mathbb{R}$ where $A\subseteq\mathbb{R}$, then if $c\in A$ is an limit point ...
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Can the irrationality of $\sqrt3$ be proven geometrically by infinite descent, similarly to Tom Apostol's proof of the irrationality of $\sqrt2$?
A geometric proof of the irrationality of $\sqrt{2}$ works by constructing two right isosceles triangles with legs $n$ and hypotenuse $m$, and finding in the construction similar triangles with legs $...
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Solving Intervals Problem Without Outer Measure
Can the following theorem on intervals be proved by elementary means, without using the outer measure [1, Chap 2] ?
Theorem
If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ with $\...
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How do I prove exponent rules for all real numbers?
It is easy to prove exponent rules for only the positive integers. For example, a^m.a^n = (a.a.a.a...)m times . (a.a.a.a.a....)n times so we just add the exponents m and n to get a^m+n. But how do I ...
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Alternative proof of existence of infinitely many positive integers not of the form $2ij+i+j$
It is relatively easy to prove that there exist infinitely many positive integers not of the form $2ij+i+j$, where $i,j\in \mathbb N$.
Sundaram already found that some positive integer is prime if and ...
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Alternative proof of uniqueness in Problem 3.6.6 of Velleman's "How to Prove it"
I am going through Vellemen's How to prove it in my free time.
Currently I am working on problem 6 in section 3.6:
Prove that there is a unique A ∈ P(U) such that for every B ∈ P(U), A ∪ B =B.
I ...
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Convergence in $L^p$ preserves density
Let $p \in [1, \infty]$. Let $(f_n)$ be a Cauchy sequence of probability density functions (p.d.f.) in $L^p (\mathbb R^d)$. Then there is $f \in L^p(\mathbb R^d)$ such that $\|f_n-f\|_{L^p} \to 0$. ...
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Prove $\mathcal T∘\mathcal R=\mathcal T∘\mathcal S$ for any $\mathcal T$ iff $\mathcal R∘\mathcal T=\mathcal S∘\mathcal T$ for any $\mathcal T$.
I am trying to understand if for two binary relations $\mathcal R$ and $\mathcal S$ the following statements are equivalent.
$\mathcal R=\mathcal S$
$\mathcal R\circ\mathcal T=\mathcal S\circ\mathcal ...
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Portemanteau lemma proof with rate of convergence
We have in Van der Vaart-Asymptotic Statistics
The proof is given as follow.
Proof. (i) $\Rightarrow$ (ii). Assume first that the distribution function of $X$ is continuous. Then condition (i) ...
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Question on a proof that a space of linear map between normed spaces is a Banach space when the arrival space is a Banach space
I have a question concerning the proof of the result stated in the title.
To prove this theorem, we consider $(f_n)_{n\in\mathbb{N}}$ a Cauchy sequence in $L(E,F)$ :
$$
\forall\varepsilon>0, \...
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Show, using presentations, that $\Bbb Z_m\times\Bbb Z_n\cong\Bbb Z_{{\rm lcm}(m,n)}\times \Bbb Z_{\gcd(m,n)} .$
Note: This is an alternative-proof question and thus is not a duplicate.
The Question:
Show, using group presentations, that $$\Bbb Z_m\times\Bbb Z_n\cong\Bbb Z_{{\rm lcm}(m,n)}\times \Bbb Z_{\gcd(m,...
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Brezis' exercise 6.21.3: prove that there is $K_2 >0$ such that $K_2 d(u, N) \le p (u)$ for all $u \in V$
I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $(V, \| \cdot \|)$ and $(H, | \cdot |)$ be real Banach spaces satisfying $V \subset H$ with compact injection. Let $p(\cdot)$ ...
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Prove that if x in A the x is not in B, when A is a subset of C and B and C are disjoint
Wanted to see if my proof of this theorem was correct. If it is not, please correct me!
Theorem. Suppose $ A \subseteq C$, and $B$ and $C$ are disjoint. Prove that if $ x \in A $ then $ x \notin B $.
...
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How to show $\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G$
I am trying to prove that
$$\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G,$$
where $G$ is the Catalan constant and $\operatorname{Li}_2(x)$ is the dilogarithm ...
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Show $\sum_{k = 1} ^n \frac{(k - 1)n^2}{n(n - k + 1)^2} = O(n^2)$
I am trying to show $\sum_{k = 1} ^n \frac{(k - 1)n^2}{n(n - k + 1)^2} = O(n^2)$. This should be easy enough, but I am stuck and can't see how to finish it off.
My solution so far:
This is the same as ...
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Hatcher example 3b.3
In Hatcher example 3B.3, we calculate the homology of $X\times S^k$. I know there is a solution by Kunneth formula
$$
H_n(X\times S^k)\cong \bigoplus\limits_{n=r+s}H_r(X)\otimes_\mathbb Z H_s(S^k)
$$
...
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Prove that $(\Sigma a^2)^5\ge27~(\Sigma a^3b^2)^2$
$\newcommand{\cyc}{\sum\limits_{\rm cyc}}$You can use \cyc for $\cyc$ in this topic.
Let $a$, $b$, $c$ be positive real numbers. Prove that
\[\left(a^2+b^2+c^2\...
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A simpler method to find the area of systems of inequalities in the complex plane
Find the area of the region of points $z$ in the complex plane such that $$\begin{align*}|z+4-4i| &\leq 4\sqrt 2\\
|z-4-4i| &\geq 4\sqrt 2.\end{align*}$$
Source: Art of Problem Solving (AoPS) ...
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