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

learn more… | top users | synonyms

0
votes
1answer
107 views

$T:X\to Y$ a linear operator, $Y$ finite-dimensional, then $\ker T$ is closed iff $T$ is continuous

I have to prove that if $T:X\to Y$ a linear operator on normed spaces, $Y$ finite-dimensional, then $\ker T$ is closed iff $T$ is continuous. On a lot of places I see a proof that looks like this: it ...
2
votes
1answer
39 views

Prove that $x\cdot y=0 \iff x=0$ or $y=0$ without cases

I have proved that given $x,y \in F$, $F$ a field, $x\cdot y=0 \iff x=0$ or $y=0$ by making cases for neither $x$ nor $y$ equals $0$ (and did a proof by contradiction) and then two cases for $x=0$ and ...
1
vote
1answer
35 views

Synthetic proof of a bisector length

In a exam the following was asked and I found a proof based on a law of cosines. Is there a pure synthetic proof without trigonometry? $ABC$ is a triangle with side lengths $CA=a,CB=b$. $D\in AB$ is ...
1
vote
0answers
32 views

Show that $[H,K,L] = A_5$, where $H,K,L$ have order $2$.

Let $H, K$ and $L$ be subgroups of order $2$ in some group $G$, and observe that the set $\{[h, k, l] : h \in H, k \in K, l \in L\}$ contains at most one nonidentity element, and so it generates a ...
0
votes
1answer
51 views

Showing that a wave equation $u_{tt}=ku_{xx}$ has solution $u=0$ if $u(x,0)=u_t(x,0)=0$

Show that a wave equation $\rho u_{tt}=Tu_{xx}$ has solution $u=0$ if $u(x,0)=u_t(x,0)=0$. Thoughts: This is easy using the general solution to wave equations ...
1
vote
2answers
40 views

Show that $[a,b] \subset Range(f)$

Question. Show that $[-\frac{1}{2},\frac{1}{2}]\subset Range(f)$, where $f(x)=\frac{x}{x^2+1}$. My proof. Let $y=\frac{x}{x^2+1}\Leftrightarrow yx^2-x+y=0.$ For real values of y, values of x must ...
1
vote
1answer
82 views

Contractions of maximal ideals in finite type $K$-algebras

In my commutative algebra class we proved the following theorem: Contractions of maximal ideals: Let $A\xrightarrow{\ \ f\ \ }B$ be a homomorphism of $K$-algebras. Suppose that $B$ is of finite ...
0
votes
0answers
17 views

Other proofs of uniqueness of interpolating polynomial

I think that one of the well known proofs is this one: Let $f:[a,b]\to\mathbb{R}$ be a function and $P_n:[a,b]\to\mathbb{R}$ be the interpolating polynomial for $f$ on $[a,b]$. Let the nodes of ...
3
votes
1answer
87 views

The value of $\gcd(2^n-1, 2^m+1)$ for $m < n$

I've seen this fact stated (or alluded to) in various places, but never proved: Let $n$ be a positive integer, let $m \in \{1,2,...,n-1\}$. Then $$\gcd(2^n-1, 2^m+1) = \begin{cases} 1 ...
1
vote
3answers
38 views

Proof of $\lim_{k\to \infty}{|x^k-x|\over { 1+|x^k-x|}}=0$

Is there an easy way to prove that $$\lim_{k\to \infty}{|x^k-x|\over { 1+|x^k-x|}}=0$$ with $x\in \mathbb R$ without using $\epsilon-\delta$ definition? Any help would be really appreciated
1
vote
1answer
36 views

Different ways of Proving the existence of Tensor Product

This is Just a curosity. Let $A$ be a commutative ring and $M,N,P$ be A-modules.I know that tensor product of $M$ and $N$ is a universal object ($ M \otimes N$,u) (where $M \otimes N$ is a $A$-module ...
2
votes
0answers
66 views

(ZF - Foundation) proves that $\log_2$ can't be infinitely iterated: Alternative proof

I think I solved Ex. (12) in Chapter I of Kunen's book. It states that ZF sans Foundation proves: For every set $X$, $$ \aleph(X) < \aleph(\mathcal{P}^3(X)), $$ where $\aleph(X):= \sup\{\alpha \in ...
1
vote
1answer
182 views

Alternative proof of the Riemann Sum Theorem using Mean Value Theorem for Integrals.

I've been reviewing proofs for a couple of calculus theorems and as I was trying to recall the proof of the Riemann Sum Theorem which uses Lower Sums and Upper Sums I came up with an idea to prove it ...
1
vote
1answer
30 views

Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one.

Let $X,Y$ be nonempty sets and $f:X\to Y$ be a function. Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one. My approach is by ...
0
votes
2answers
26 views

Clarification: Prove there exists a number $N$ such that $n > N$ implies $s_n >a$

Below is the proof that I have been working on and the solution provided by the professor. Let $(s_n)$ be a convergent sequence, and suppose $\lim s_n > a$. Prove there exists a number $N$ ...
0
votes
1answer
23 views

Find a criterion for divisibility

Find a criterion such that $\displaystyle\sum_{i=1}^ni$ divides $\displaystyle\prod_{i=1}^ni^2$ for $n\in\mathbb N$. What I have done so far, $\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}{2}$ and ...
5
votes
1answer
68 views

Proving that and how $ \frac{1}{n}\sum\limits_{p\le n}\lfloor n/p \rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
2
votes
1answer
46 views

Cantor's diagonal argument modified version

I have the following doubt regarding Cantor's diagonal argument. First of all, the "usual case" is quite clear for me. If $X$ is some set, then we can show there is no surjection from $X$ onto the set ...
1
vote
2answers
71 views

Prove that no set can contain everything (or every other set)

Prove that there cannot exist a set that contains everything. Ill put my proof in the answer so please check it there. Also if there is a more creative way to do this(using the basic axioms) if it's ...
1
vote
1answer
36 views

An alternative method to find $\sum_{k=1}^{2n-1} | \beta ^k - 1|$

Let $\beta \in \mathbb{C}$ such that $\beta ^n = 1$ but $ \beta ^k \neq 1$, $\forall k=1,2,\cdots, n-1$. Find the value of $$ \sum_{k=1}^{2n-1} | \beta ^k - 1|.$$ I came across such a question ...
0
votes
1answer
74 views

How to prove triangle inequality in How to Prove It Sec. 3.5 Question 12c?

(a) Prove that for all real numbers $a$ and $b$, $$|a| \le b \text{ iff } -b \le a \le b.$$ (b) Prove that for any real number $x$, $$-|x| \le x \le |x|.$$ (Hint: Use part (a).) (c) Prove that ...
3
votes
3answers
105 views

Proving uniqueness of solutions to $\sin^2A + \sin^2B = \sin (A+B)$ without using multivariable calculus

In the course of solving a trigonometric problem (see $a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle), in one approach the following equation ...
3
votes
2answers
81 views

$\mathbf{Set} \not \simeq \mathbf{Set}^*$ by considering $\{1, 2 \} \to \{1\}$

This answer gives a nice way of seeing why the category of sets is not isomorphic to its dual. I would like to know whether there is a proof from a certain different direction. When considering the ...
1
vote
1answer
54 views

Proof that $A\cap\emptyset=\emptyset$

I'm trying to prove $A\cap\emptyset=\emptyset$. I've seen several proofs for this which all seemed to essentially go about proving it by noticing that $\emptyset\subset A\cap\emptyset$ by definition ...
0
votes
1answer
52 views

About a proof regarding a property of groups of order $pq$ where $p$ and $q$ are primes

I'm studying right now Automorphisms in Dummit & Foote's Abstract Algebra (Section 4.4). In pages 135-136, the following example is given: and here's Proposition 16 muntionned in the ...
5
votes
0answers
83 views

Showing that only $(n+1)^{n-1}$ of all the possible $n^n$ choices assure a full car park

This exercise is taken from the site of Queen Mary University of London: A car park has $n$ spaces, numbered from $1$ to $n$, arranged in a row. $n$ drivers each independently choose a favourite ...
1
vote
0answers
39 views

Uniform Continuity of $\frac {1}{x}$ on [$a, \infty$) for positive $a$

$\frac {1}{x}$ behaves nicely in that it's monotone and the derivative is monotone also. So on [$a,\infty$] it can be seen that the $\delta$ which will work everywhere is the $\delta_1$ at the end ...
0
votes
1answer
112 views

Is this alternative proof of Theorem 3.7 (“Baby” Rudin, Ch. 3) correct and, if so, well written?

Rudin, in his Principles of Mathematical Analysis, proves the following theorem: The subsequential limits of a sequence $\{p_n\}$ in a metric space $X$ form a closed subset of $X$. I've tried to ...
4
votes
5answers
297 views

How to prove $3^\pi>\pi^3$ using algebra or geometry?

It's a question of a some time ago test, I've found a way to solve the problem using calculus, but always I've thought that exist a solution with algebra and geometry. Thank you for your time.
3
votes
0answers
58 views

Proof that 10 lines pass through the centroid of a triangle

Let $A$, $B$, $C$, $D$, and $E$ be points on a circle. For any three points, we draw the line going through the centroid of the triangle formed by these three points that is perpendicular to the line ...
0
votes
2answers
52 views

Verifying a Proof for Spivak's Calculus Question (Chapter 2 Problem 9)

It says "Prove that if a set $A$ of natural numbers contains $n_0$ and contains $k+1$ whenever it contains $k$, then A contains all natural numbers $\ge n_0$". Am I allowed to construct another set ...
2
votes
2answers
111 views

prove if f(x) has an infinite limit then limit of 1/f(x) is = 0

I wanted to ask if someone can do me the favor pointing out the mistakes I might of made in proving the theorem below. Also is there a way to prove the theorem without using the definition of limits? ...
-1
votes
2answers
42 views

Confusion with the reconstruction conjecture?

After reading about the reconstruction conjecture for graphs, I came up with what I thought was a proof by contradiction. Consider the class $T$ of (isomorphism classes of) finite graphs, and the ...
0
votes
3answers
73 views

How to prove a specific sequence is Fibonacci's with no prior knowledge nor trial and error?

Let $n$ be a positive integer and let $s_n$ be the number of increasing sequences of integers, alternatingly even and odd, starting with $0$ and ending with $n$. E.g. for $n=3$ we only have the two ...
2
votes
3answers
119 views

Another identity with binomial coefficients

I'm looking for an easy way to prove this identity $$\sum_{j=0}^{n}{(-1)^j j (n-j) {n \choose j}} = 0$$ for $n > 2$. I know this can be proven by differentiating $(1+x)^n = \sum x^j{n \choose j}$ ...
1
vote
0answers
33 views

Rational analogue of expansion to base b

As is well known, we can expand every positive integer $n$ to a base $b \in \Bbb N$ in the form $$n = \sum_i a_ib^i ,\ \ \ 0\leq a_i \leq b_i-1$$ uniquely. Less well known is that we can do this for ...
12
votes
1answer
326 views

Alternative proof of simple integral inequality

Problem. Let $f\in C^1(\mathbb R)$ such that $f(0) = 0$ and $0 < f'(x) \le 1$. Prove that for all $x\ge 0$ $$ \int_0^x f^3(t)\,dt \le \left(\int_0^x f(t)\,dt\right)^{\!\!2}. $$ Below is my ...
0
votes
1answer
60 views

Minimizing the area of the triangles containing a square of side $1$

This exercise is from a past admission exam to an Italian institute: Among all the triangles that contain a square of side $1$, which ones have minimum area? I have solved it, however I'd like ...
6
votes
5answers
210 views

$p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.

I want to prove that for $p,q,r$ different primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational. Is the following proof correct? If $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is rational, then ...
0
votes
0answers
34 views

(Theoritical Question) Multiply two negatives, get a positive, but… [duplicate]

Many books explain it because you can solve it with addition. I'm interested, is there another way to prove it?
1
vote
1answer
135 views

Prove that “No one likes Reggae music” is the same as “Everyone does not like Reggae music”.

I interpreted this as a case of the extension of De Morgan's Law to quantifiers. https://en.wikipedia.org/wiki/De_Morgan%27s_laws#Extensions I know that similar questions have been asked before about ...
0
votes
4answers
178 views

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of ...
1
vote
0answers
87 views

IMC 2014, Problem 4 [Day 2]

We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points ...
3
votes
1answer
68 views

How to resolve the issue of two sequences converging to zero for $n, m \to \infty$?

My question is motivated by the following exercise in probability theory: Let $X_n \to X$ in probability and $X_n \geq Y$ a.s. Show that $X \geq Y$ a.s. I noticed that for all $n, m \in ...
3
votes
1answer
88 views

Solving the trigonometric equation $\tan^2x+\cot^2x=2-\cos^{2014}(2x)$

I was solving the trigonometric equation $$\tan^2x+\cot^2x=2-\cos^{2014}(2x) $$ I solve it by inequality $|a|+\frac{1}{|a| }\geq 2$. $$ L.H.S=\tan^2x+\cot^2x =\tan^2x+\frac{1}{\tan^2x} ...
2
votes
0answers
77 views

Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
7
votes
3answers
266 views

Abel-Ruffini theorem, Galois theory and minima and maxima

Questions: Does there exist a proof of the Abel-Ruffini theorem without using Galois theory? Does there exist a proof that there exists a polynomial $P$ with $\deg P = 5$ such that the roots are not ...
0
votes
2answers
93 views

Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$.

I have been working through the following proof: Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$. Below, I have included screenshots of the ...
0
votes
3answers
68 views

Help with proof: $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$

The question is: If $A,B$ are any $m\times n$ matrices, prove that $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$. ($\mathrm{rank}(A)$ is the dimension of the column space of $A$, ...
3
votes
4answers
279 views

$S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$.

Here is the problem that I am currently working on: $S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$. I have access to the answer for this proof, and wanted help with the first ...