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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Showing $ \left \lceil \frac{a}{b} \right \rceil \leq \frac{a+b-1}{b}$ [duplicate]
I'm trying to convince myself how one can hypothesize that
$$ \left \lceil \frac{a}{b} \right \rceil \leq \frac{a+b-1}{b}$$
or alternatively that
$$ \left \lceil \frac{a}{b} \right \rceil -\frac{a}{b}\...
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Different Characterization of Basis of a Vector Space
Definition: $(1)$ $B$ is called maximal linearly independent set if $B$ is independent and if $S\supset B$, then $S$ is dependent, or, equivalently, $\nexists S\supset B$ such that $S$ is independent. ...
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There are only finitely many integer lattices with bounded covolume
I have an uninsightful proof for the following lemma (I discuss motivation below). Let $C>0$ and $n\geq 1$ be fixed, set $\def\Z{\mathbb{Z}}M=\Z^n$.
Lemma. There are only finitely many subgroups $...
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Deriving the formula for the number of distinct de Bruijn sequences
I am reading both https://en.wikipedia.org/wiki/De_Bruijn_sequence and the original paper by Bruijin himself
How to derive step-by-step the formula for the number of distinct de Bruijn sequences since ...
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Theorem 6, Section 2.3 of Hoffman and Kunze’s Linear Algebra
If $W_1$ and $W_2$ are finite-dimensional subspaces of a vector space $V$, then $W_1+W_2$ is finite-dimensional and $\mathrm{dim}(W_1)+ \mathrm{dim}(W_2)= \mathrm{dim}(W_1\cap W_2)+ \mathrm{dim}(W_1+...
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Alternate proof that infinite complete binary branching tree as a Kripke frame generates the S4-tautologies.
The $\mathsf{S4}$ tautologies are precisely the formulas of modal logic that hold in all transitive, reflexive Kripke frames.
I think I've found an example of a single Kripke frame, the infinite ...
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(Generalized) Cantor set is uncountable
The standard Cantor set formed by recursively removing the middle one-thirds, on the interval $[0, 1]$ can be shown to be equal to the uncountable set of the base-3 numbers between $0$ and $1$ with ...
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Alternative Solution to Theorem 5 Corollary 2, Section 2.3 of Hoffman’s Linear Algebra
In this video lecture, time stamp 19:20 - 24:30. Professor show proof of following claim: (Extend to basis) Every linearly independent list of vectors in a finite dimensional vector space can be ...
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Ways to show $\int_0^{\infty}\frac{\sin^2(\pi x)}{x^2}\Big\lvert x-\Big\lfloor x +\frac12 \Big\rfloor \Big\rvert \, \mathrm{d}x = \frac{\pi^2}{8}$?
Whilst reading about Lobachevsky's integral formula I tried constructing some interesting integrals which could be evaluated with said formula. One result I found was
$$\int\limits_{0}^{\infty} \frac{...
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Closed form for $\sum_{k=1}^\infty\frac{H_k^{(m)}}{k^n}$
Let's define
$$\sigma(m,n)=\sum_{k=1}^\infty\frac{H_k^{(m)}}{k^n}$$
where $H_k^{(m)}=\sum_{n=1}^{k}\frac{1}{n^m}$ is the k-th generalized harmonic number of order $m$.
In mathworld site Eq (20), I ...
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What are some elementary ways to show that there's no solution to $\pm1 \pm 2 \pm 3\dots \pm n =0$ for $n \equiv 1,2 \ \, (\bmod 4\,)$?
I was looking at sequence A063865 of the OEIS, which is given by the number of ways of choosing $+$ and $-$ signs such that $\pm1 \pm 2 \pm 3\dots \pm n =0$. If we define the $n$-th element of this ...
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Given a sequence with $a_1=1$ and $a_{n+1}=a_n-\frac13a_n^2$. Is there an easier way to get an upper bound of $1/a_{100}$?
Suppose that a sequence $\{a_n\}_{n=1}^\infty$ satisfies $a_1=1$ and $$a_{n+1}=a_n-\frac13a_n^2,\qquad n=1,2,3,\cdots.$$
Which of the following is true?
(A) $2<100a_{100}<\frac52$
(B) $\frac52&...
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Correctness of method - if $\{ y_n \} $ is unbounded, prove $\{ y_n z_n\} $ does not converge to $0$
Problem: Let $c_0$ be the space of complex sequences that converge to $0$. Let $\{ y_n\}_{n \in \mathbb{N}} $ be a sequence of scalars such that for arbitrary sequence $\{ z_n\} \in c_0 $ we have: $...
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Is this alternative proof that $\sqrt 2$ is irrational sound?
I have just taught the classic proof by contradiction that $\sqrt 2$ is irrational, and one of my students came up with the following proof:
Assume that $\sqrt 2$ is rational, so $\sqrt 2=\frac{a}{b}$ ...
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Is there a simplified 4-color proof for pixilated maps?
In mathematics, the four color theorem, or the four color map theorem,
states that no more than four colors are required to color the regions
of any map so that no two adjacent regions have the same ...
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Proof about concave decreasing function with some constraints
Hi it's a follow up of How to prove that the functional equation $f(x)+f(y)=f(\frac{xf'(x)+yf'(y)}{f'(x)+f'(y)})+f(\frac{x+y}2)$ is verified only by some basic functions? :
Conjecture :...
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Evaluating $\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx$
How to show that
$$\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx=\frac{5\pi^3}{64}+\frac{\pi}{16}\ln^2(2)-4\,\text{G}\ln(2)$$
without breaking up the integrand since we already know:
$$\int_0^1\...
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Looking for alternative proof of one-to-one correspondence b/n the real interval (0, 1] and the nonterminating decimal expansions 0.d1d2d3…
As per Michael J. Schramm, 1996, Introduction to real analysis, Theorem 6.6:
In an Archimedean ordered field in which the Nested Intervals property holds, there is a one-to-one correspondence between ...
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For which values $k \in \mathbb{R}$ does $kz - \tan z$ have non-real roots?
This is not a homework or university course question, it is one purely of my own posing that arose while investigating the Weierstraß factorization of various complex functions.
Numerical Examples ...
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The Converse of the Radon-Nikodym Theorem for Signed Measures
I noticed that on Wikipedia there is a statement of Radon-Nikodym Theorem for signed measures: https://en.wikipedia.org/wiki/Radon–Nikodym_theorem#For_signed_and_complex_measures. I would like to see ...
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(Dis)prove $f(2\sqrt{xy}+\frac{1}{3xy}(\exp(\frac{\ln(xy(x-y)^{2}+1)}{xy(x-y)^{2}+1})-1))-\frac{xy}{1-x-y}\ge 0$
I would like to show if true :
Let us consider the inequality for $x,y>0$ and $x+y\leq 0.5$ :
$$G\left(x,y\right)=f\left(2\sqrt{xy}+\frac{1}{3xy}\left(\exp\left(\frac{\ln\left(xy\left(x-y\right)^{...
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$f$ is a polynomial and $f(0)\neq0$. Why is $fg+x$ a perfect square for some $g$?
On Hacker News, someone posted the following exercise: prove that every nonsingular complex symmetric matrix $M$ has a symmetric matrix square root.
This is old chestnut. As the poster indicated, ...
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is proof by contradiction essential for proving the Pigeonhole Principle? [duplicate]
The standard way to prove the Pigeonhole Principle that, putting $n+1$ pigeons into $n$ pigeonholes necessitates the existence of at least one pigeonhole with at least 2 pigeons is by contradiction. ...
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Finding Subspaces $U, V, W < \mathbb{R}^4$ such that $\mathbb{R}^4 = U \bigoplus V = U \bigoplus W = V \bigoplus W$
So, the dimension of each of the subspace must be equal to 2. Now, I kind of brute-forced this problem by trying different values and came up with $$ U = Lin((1, 0, 1, 0), (0, 1, 0, 1))$$ $$W = Lin((1,...
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At least 3 points on the convex hull
It's almost obvious that for any point set that contains at least 3 points on the Euclidean plane the boundary of its convex hull must contain at least 3 of the points. I worke out a proof (not sure ...
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Quick proof: orthogonal matrix commuting with special orthogonal matrix is special orthogonal
Is there a quick way to prove the following statement?
Let $M\in\mathrm{O}(2k)$ and $P\in\mathrm{SO}(2k)$. Suppose $P$ has no real eigenvalues and commutes with $M$. Then $\det M=1$, i.e., $M\in\...
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Alternate proof that $\cos(3A)+\cos(3B)+\cos(3C)=1-4\sin(\frac{3A}2)\sin(\frac{3B}2)\sin(\frac{3C}2)$ for $A$, $B$, $C$ the angles of a triangle
$A$, $B$, $C$ being the angles of a triangle, we need to prove that:
$$\cos(3A)+\cos(3B)+\cos(3C)=1-4\sin\left(\frac{3A}2\right)\sin\left(\frac{3B}2\right)\sin\left(\frac{3C}2\right)\tag*{}$$
The ...
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Show the inequality $\frac{\sqrt{\pi}}{2}<\left(\pi-e\right)!$
Working a bit on
About the inequality conjectured as $x!>\left(\arctan\left(\cosh\left(x\right)\right)\right)^{a}$ for $x>0$ and fixed $a$
I got the inequality:
$$\frac{\sqrt{\pi}}{2}<\left(\...
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Does the question that every group of order 77 must have an element of order 7 and an element of order 11 have an elementary proof
Question 21 of Chapter 10 of Abstract Algebra by Dan Saracino is as follows:
Prove that every group of order 77 must have an element of order 7 and an element of order 11.
The more general case of ...
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Exercise 4, Section 32 of Munkres’ Topology
Show that every regular Lindelöf space is normal.
My attempt: Let $A$,$B$ be closed in $X$ such that $A \cap B=\emptyset$. By chapter 7 theorem 2.2 of Dugundji topology (equivalent definition of $T_3$...
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Examples of Intuitionistic Proof
For uni I need to write a paper and give a presentation on intuitionism, and I am looking for nice examples of theorems or other results to prove or disprove intuitionistically. One example I found ...
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Is there a shorter or more trivial way to prove that $ x > \cos (x)-\cos (2 x) $ holds for all $x>0$?
I want to prove that the inequality
$$ x > \cos (x)-\cos (2 x) $$
holds for all $x>0$.
My attempt:
Since the function on the RHS is periodic, we can find the position of extrema (on first ...
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Possible ways of solving an Olympiad question
"A piece of land of a square shape with dimensions 10m x 10m is divided into 100 square parcels with dimensions 1m x 1m. Initially, 9 of the parcels are overgrown by weed. If a parcel is ...
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There exists a ball inside the convex hull of the union of $2$ other balls
I'm trying to show that given a point in the line segment connecting centers of $2$ given balls, there is a ball centered at that point inside the convex hull of the union of that $2$ other balls.
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Help me rewrite Theorem 3 from Serge Lang's Basic Mathematics using Velleman's 'given–goal diagrams'
I want to apply the proof strategies and overall scratch work diagram based framework introduced in the book "How to Prove It" in order to rewrite the following theorem:
Theorem 3. Any ...
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A square on the equator of a sphere is a critical point of the electrostatic potential
$\newcommand{\S}{\mathbb{S}^2}$
This is a self-answered question. I learned something from spelling out the details, and I hope this could be interesting to others. I would welcome alternative ...
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The intuition behind this explicit form of an affine hull?
I have come across an explicit formula of an affine hull from here. I'm trying to prove that formula in 2. below. However, my approach is quite muscular and not natural, i.e. I have to know the ...
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Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$?
here's my question
If $y^2=ax^2+bx+c$ then prove that $y^3[d^2y/dx^2]$ is a constant .
I have solved this using the conventional method, taking square root, differentiating w.r.t to x and using ...
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Question on analytic continuation of the principal branch of logarithms on $B(1, 1) $
(Excercise $6$ ,Conway ,page $217$ )
Let $D_0=B(1, 1) $ and $f_0$ be the restriction to $D_0$ of the principal branch of logarithm. For $n\in \Bbb{Z}$ let $\gamma(t) =e^{2\pi i nt}$ for $0\le t\le 1$ ....
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Let $a,b,c \le 0$. Then $\max(a,c)+\max(b,c) \le \max(a+b, c)$
I'm reading a lecture note in which the author uses
Let $a,b,c \le 0$. Then
$$
\max(a,c)+\max(b,c) \le \max(a+b, c).
$$
Below is my attempt. Is there other way to look at the problem and have a more ...
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Showing that $\{S^x_{\tau \wedge n}\}_{n\in\mathbb{N}}$ is uniformly integrable
Let $\{X_n\}_{n\geqslant 1}$ a sequence of i.i.d. Rademacher r.v., that is $\Pr [X_1=1]=\Pr [X_1=-1]=\frac1{2}$ and for $n,m\in \mathbb{Z}$ with $n<m$ set $X_0:\equiv x$ for some $x\in(n,m)\cap \...
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Exercise 7(b), Section 31 of Munkres’ Topology
Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1} \big( \{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (b) Show that if $X$ is ...
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Exercise 7(a), Section 31 of Munkres’ Topology
Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (a) Show that if $X$ is ...
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Exercise 6, Section 31 of Munkres’ Topology
Let $p \colon X \to Y$ be a closed continuous surjective map. Show that if $X$ is normal, then so is $Y$. [Hint: If $U$ is an open set containing $p^{-1}(\{ y \} )$, show there is a neighborhood $W$ ...
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Proof for weighted sample central moments
I think the question is valuable for the community as I could not find out any close related topic on stackexchange or even internet.
We have a sample of observations and we decided to replicate some ...
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Is it valid/useful to prove statement $X$ by finding $Y$ such that $Y\to X$ and $\lnot Y\to X$?
Background: Suppose I want to prove theorem X. Typically, I'd have to use a set of axioms $A = \{A_1,A_2, \ldots ,A_n\}$ or previously proved theorems $T=\{T_1,T_2, \ldots,T_n\}$ and consider all of ...
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Suppose $S$ is linearly independent set of vectors, $T\subseteq S$, then $T$ is linearly independent (alternate method)
I know this question has answers here but they involve linear transformations which I haven't reached upto yet. There was a proof by contradiction there as well but I had a different idea in mind:
Let ...
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Showing that K-topology is not path-connected without using compactness
Let $\mathbb{R}_K$ denote the real line in the K-topology, which is the topology generated by the basis $\left\{(a,b)|a,b \in \mathbb{R}\right\} \cup \left\{(a,b)-K|a,b \in \mathbb{R}\right\}$, where $...
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Exercise 1, Section 32 of Munkres’ Topology
Show that a closed subspace of a normal space is normal.
My attempt:
Approach(1): Let $\{y\}$ be a singleton set in $Y$. Since $X$ is $T_1$, $\{y\}$ is closed in $X$. $\{y\} =Y\cap \{y\}$. By theorem ...
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Show that $SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$
Show that $SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$
I know that $|SL_2(F_3)/Z(SL_2(F_3))|= 12$.
Then if the quotient group has a normal subgroup of order $4$ then it is isomorphic to $A_4$.
Suppose that it ...
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