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
0answers
37 views

proof that $\frac{e^{-t}}{2}(t^2+2t+2)\le1$ for $t\ge0$

Show that $\forall t\ge0,x\le1$ where $$x=\frac{e^{-t}}{2}(t^2+2t+2),t\in\mathbb{R}.$$ My proof we have $x=\frac{(t^2+2t+2)e^{-t}}{2}$ then ...
9
votes
3answers
1k views

Prove that there is no smallest positive real number

I have to prove the following: $$\text{Prove that there is no smallest positive real number}$$ Argument by contradiction Suppose there is a smallest positive real number. Let $x$ be the smallest ...
0
votes
1answer
450 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 ...
2
votes
0answers
27 views

Product Spaces: Tube Lemma

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. My professor asked to prove the following Lemma. The Tube Lemma: Let $K$ be a compact ...
1
vote
1answer
96 views

Cutting chocolate diagonally

Given is chocolate with rectangular pieces of size $a \times b$. If cut diagonally, how many pieces will it be split into? If knife passes exactly by co-catenating we assume there is no damage to ...
0
votes
0answers
32 views

Determining if a rational number has a terminating decimal expansion (proof)

Theorem: $x=\frac pq$ is any given rational number, $n$ and $m$ are any whole numbers (including zero) which you can choose. a) If $q=2^n5^m$ is possible, $x$ has a terminating decimal expansion. ...
1
vote
0answers
219 views

Is independence a transitive property? [duplicate]

If the events $A$ and $B$ are independent and the events $B$ and $C$ are independent, does this necessarily mean events $A$ and $C$ are independent? I used coin tosses to try to model this with $A = ...
1
vote
1answer
37 views

Dijkstra's Algorithm for Negative Weights.

Now the problem states that their is a graph $ G = (V,E) $ where some of the edges have negative weights while some of the edges have positive edges. Now the question is why won't Dijkstra's algorithm ...
3
votes
3answers
95 views

Can I find $\sum_{k=1}^n\frac{1}{k(k+1)}$ without using telescoping series method?

I am interested to find $$\sum_{k=1}^n\frac{1}{k(k+1)}$$ without using telescoping series method. I tried very hard but still could not think of a way to find it without using telescoping series ...
2
votes
8answers
335 views

A pedagogical proof that 9's can be ignored when calculating digital roots

I was asked by an elementary school teacher for a proof that you can ignore all 9's when calculating the digital root of a number. For instance, when calculating the digital root of 7593329, you ...
4
votes
3answers
304 views

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number My attempt: Consider the polynomial $ (x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0 $. By the rational root theorem, we can conclude that $ \pm ...
19
votes
1answer
217 views

$\log_2 13$ is irrational

Is it true that $\log_2 13$ is irrational? Let $x=\log_2 13\implies 2^x=13$. 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, ...
0
votes
0answers
26 views

Proving associativity of symmetric set difference

I'm proving that $P(X)$ (the set of the subsets of $X$) is a ring with the following operations: If $A, B \subset X$, then $A+B := (A \cup B) \backslash (A \cap B) $ and $A \cdot B = A \cap B $. I ...
5
votes
1answer
53 views

$\mathbb{Z}[\sqrt{2}]/(3-\sqrt{2})$ is ring isomorphic to $\mathbb{Z}_n$.

what would be an $n$ such that $\mathbb{Z}[\sqrt{2}]/(3-\sqrt{2})$ is ring isomorphic to $\mathbb{Z}_n$? This problem was on a qualification test. Here's how I solved it, but I'm not satisfied ...
0
votes
0answers
22 views

Lebesgue Measurable By Alternative definition of Measure

Prove that any compact set $K$ in $R^{n}$ is Lebesgue measurable and $m(K) < \infty$ Actually the proof of this is given in Stein and Shakarchi's book on Real Analysis (Page 38, Property 4) where ...
0
votes
1answer
648 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 ...
2
votes
0answers
55 views

Where can I find Wielandt's original proof of Sylow's Theorem?

I have seen several proofs of Sylow's Theorem based on Wielandt's method. Everyone gives credit to Wielandt's proof of Sylow's theorem, but ironically everyone puts their own spin on it. Where can I ...
16
votes
3answers
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
votes
2answers
60 views

Proof of irrationality without using contradiction

I'm just wondering if there exists proofs that certain numbers are irrational that do not begin by saying some like along the lines of "assume $k=a/b$ for integers $a$ and $b$" and then deduce a ...
2
votes
0answers
47 views

How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
2
votes
2answers
88 views

Proving $2^{2^n}+3^{2^n}+5^{2^n}$ is divisible by $19$ for all $n\geq 1$ by induction

I came across the following in the book Handbook of Mathematical Induction: $$ 19\mid (2^{2^n}+3^{2^n}+5^{2^n}),\quad n\in\mathbb{Z^+}\tag{1} $$ Apparently, this problem is not so bad if you think ...
0
votes
4answers
53 views

If $x>y$, then $x \bmod y < \frac{x}{2}$

Given two natural numbers $x$ and $y$ such that $x > y$, prove that $$x \bmod y < \frac{x}{2}.$$ I was planning on solving this proof by using the definition of mod; however, I was wondering ...
3
votes
4answers
828 views

Axiom of Choice and Right Inverse

I read an Theorem that states: Let $A$ and $B$ be non-empty sets, and let $f:A \to B$ be a function, then the function $f$ has a right inverse if and only if $f$ is surjective. The Theorem ...
0
votes
0answers
17 views

Proof Method and Absolute Value

I'm interested in the following method of proof for inequalities involving modulus : $( -x \geq C \wedge x \geq C ) \to ( |x| \geq C )$ $( -x > C \wedge x > C ) \to ( |x| ...
0
votes
0answers
41 views

coloring theorem for topological partitions

Let $(X,\tau)$ be a topological space. DEFINITIONS: Define a topological partition of $X$ into connected sets to be a collection of pairwise disjoint open connected sets $\{U_i\}_{i\in I}$ such that ...
1
vote
2answers
47 views

Proof that real multiplication distributes over addition using Dedekind cuts?

Proving $\forall x,y,z\in \mathbb R,\:x(y+z)=xy+xz$ There is a very concise proof of this in ProofWiki using Cauchy sequences, but I was wondering whether the same would be possible using Dedekind ...
2
votes
2answers
71 views

Is there a much simpler proof for Euler factorial formula?

Euler formula for factorial stated as follows Theorem [Euler]: For any non negative integers $a$ and $n$ such that $a\geq n$ $$ n!=\sum_{k=0}^{n}(-1)^k\binom{n}{k}(a-k)^n$$ Proving this ...
2
votes
2answers
47 views

determine the values that series converges

Determine for what values of $x \in \Bbb R$ the series $$\sum_{n = 1}^\infty \frac{(-1)^n}{2n+1}\left(\frac{1-x}{1+x}\right)^n$$ coverges. I have tried the alternating series test but I don't think ...
1
vote
0answers
26 views

How to show that a function is continuous in the topology of weak convergence

Let $\Omega$ be compact, and let $\omega^* \in \Omega$ be arbitrary. Let $\Delta (\Omega)$ denote the set of all probability measures over $\Omega$, and endow $\Delta ( \Omega)$ with the topology of ...
3
votes
1answer
131 views

Is there a memorable solution to Kirkman's School Girl Problem?

Given a solution to Kirkman's School Girl Problem, it is of course easy enough to check that it actually is a solution. But how could you reconstruct it if you lost it? Is there a method or algorithm ...
4
votes
0answers
88 views

${n \choose r}=\frac {n!}{r!(n-r)!}$ without using the permutation approach.

I had an idea that would be to first prove Pascal's Rule, $${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r},$$ which can be proved combinatorically whether one particular element (among the $n$) ...
8
votes
5answers
1k 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? ...
2
votes
1answer
54 views

Geometric proof of this property of the ellipse

I came across the following property of the ellipse: The distance from a focus of an ellipse to any point on the ellipse is equal to $a(1-e \cos\theta)$. Where the $a$ is the length of ...
2
votes
1answer
28 views

adjoint of an operator. on $L^2(0,1)$, $Bf(x)=\int_0^x f(t)dt$

I see that the above operator is bounded. I ended up with an argument to calculate the adjoint as follows, $$ <f,Bg>=\int_0^1\overline{f(x)} \int_0^xg(t)\,dt\,dx $$ I see $f(x)$ as the ...
2
votes
1answer
27 views

Consider a linear operator $L$ and some polynomial of it, $L'=p(L)$. Show that the minimal polynomial of $L'$ has smaller degree than that of $L$.

Consider a linear operator $L$ and some polynomial of it, $L'=p(L)$. Show that the minimal polynomial of $L'$ has degree less than or equal to the minimal polynomial of $L$. First, start working over ...
2
votes
0answers
88 views

Alternative proof that Harmonic sum is not an integer

In the post “Is there an elementary proof that ∑k=1/k is never an integer?” there is a simple and elegant proof by Bill Dubuque, who uses the prime 2 as a basis for his proof. I wondered whether a ...
3
votes
2answers
117 views

Prove: if $f(x) =x\sin (\pi x)$ then $f'(x)$ vanishes at a unique point in $ ( n + 1/2, n) $

Let $ f(x) = x\sin (\pi x), x > 0 $. Then prove that for all natural numbers n, $f'(x)$ vanishes at a unique point in $ ( n + 1/2, n) $ The given solution shows a graph, but is there any algebraic ...
4
votes
3answers
139 views

Proof of no primes such that $x^2 + y^2 = z^2$ [duplicate]

I'm in a pretty simple "CS Math" course for year 1 Comp Sci, and I came across this: Disprove, $x^2 + y^2 = z^2$, such that $x, y, z$ are primes I thought of this as, if n is a prime, then prime ...
19
votes
4answers
2k views

Proofs that every natural number is a sum of four squares.

I am planning to write a little note detailing several proofs of Lagrange's theorem that every natural number can be written as the sum of four perfect squares. I know of three different proofs so ...
0
votes
0answers
50 views

$g$ o $f$ is nowhere continuous

Find an example of a function $f$ discontinuous on $\mathbb{Q}$ and another function $g$ discontinuos at only one point, but $g$ o $f$ is nowhere continuous. One solution is: Let $f$ be the ...
1
vote
0answers
26 views

Solutions to the heat equation given certain growth conditions

Let $u(x, t)$ be the solution of the following Cauchy problem for the heat equation given by \begin{align*} u_{t}(x, t) - u_{xx}(x, t) &= 0 \quad (x, t) \in \mathbb{R} \times (0, T)\\ u(x, 0) ...
7
votes
1answer
157 views

Stefan-Boltzmann Constant and Stefan's Law

The following argument is from my textbook, An Introduction to Thermal Physics by Daniel Schroeder. If you are familiar with the derivation of Stefan's Law from the energy density of a photon gas, you ...
1
vote
0answers
166 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?
3
votes
1answer
55 views

Prove the operator is positive

I'm searching for an alternative proof of the following: Let $U$ be a self-adjoint operator on a Hilbert space $H$, define $m=\inf_{\|x\|=1}\langle Ux,x\rangle$ and $M=\sup_{\|x\|=1}\langle ...
6
votes
1answer
86 views

Is $\int f=f-1\iff f(\cdot)=e^{\cdot}$ proved this way correct?

I saw this on math overflow and made me wonder, why does it work, is it rigorous, can we really factor like this, and where can we use similar tricks; Let $\int$ denote $\int_0^x$ Then solve $$\int ...
5
votes
2answers
41 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
1
vote
1answer
138 views

Proving the Sine Rule with one line.

Working on a general proof of the Law of Sines for ALL Euclidean triangles. Right triangles are easy. Acute triangles are just two proofs of the right triangle. But this is not sufficient for me. I ...
0
votes
0answers
30 views

Proof: The reduced row echelon form of a matrix is unique.

If $A \in M_{m\times n}$ with real entries, then there exist a unique matrix $R$ in row echelon form such that $A\sim R$, where $R$ comes from $A$ after performing elementary operations. How can I do ...
2
votes
2answers
114 views

$X$ compact Hausdorff space, characterize the maximal ideals of $C(X)$

I know this question has been asked before, but I think I'm very close to a new solution and wanted to know if it is a viable approach. Let $C(X)$ be the ring of continuous functions $X \rightarrow ...
2
votes
10answers
144 views

How to prove $x^3-y^3 = (x-y)(x^2+xy+y^2)$ without expand the right side?

I can prove that $x^3-y^3 = (x-y)(x^2+xy+y^2)$ by expanding the right side. $x^3-y^3 = (x-y)x^2 + (x-y)(xy) + (x-y)y^2$ $\implies x^3 - x^2y + x^2y -xy^2 + xy^2 - y^3$ $\implies x^3 - y^3$ I was ...