1
vote
2answers
59 views

How to prove that the sequence is decreasing $a_{n}=\frac{ln(n)}{n^2}$

Is my way/proof good and completely mathematically rigorous? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
1
vote
2answers
57 views

Prove that n!+1 contains a prime factor greater than n and use this to prove that there are infinte many primes [duplicate]

Prove that $n!+1$ contains a prime factor greater than $n$ and use this to prove that there are infinitely many primes. I said assume that $n!+1$ contains a prime $p$ which is less than or equal to ...
2
votes
1answer
80 views

Prove that the dual graph of any (planar) graph is connected

I'd like to know if there's a standard proof that the dual graph of any planar graph is connected (or, if there's a counterexample, I'd like to know that too). I've thought of a proof that might work ...
0
votes
0answers
25 views

A compendium of proof-techniques per objective

Please consider this as an on-going list of techniques preferably per objective or subject. Many mathematical books (at least lately) are focusing on "design patterns" if you like of proof-techniques ...
0
votes
2answers
30 views

Different way showing a subgroup is a subgroup of another subgroup

http://crazyproject.wordpress.com/2010/04/11/subgroups-and-quotient-groups-of-solvable-groups-are-solvable/ Lemma 1: Let $G$ be a group and let $H,K,N \leq G$ with $N$ normal in $H$. Then $N \cap K$ ...
1
vote
2answers
36 views

Easier Proof - Union of finite lin-indep subsets of the eigenspaces = a lin-indep subset. [Lay P285 Thm 5.3.7c]

P267 Lemma. Let $T$ be a linear opera $tor$, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. For eacb $i=1,2,\ \ldots,\ k$, let $v_{i}\in E_{\lambda;}$, the ...
0
votes
1answer
35 views

Adaptation of this proof of spectral theorem to the complex case

My question is quite simple, I would like to know why we can't use this proof to the complex case, i.e., the operator $T$ is self adjoint on a complex n-dimensional inner product space $V$. Can we ...
0
votes
0answers
41 views

Mathematical logic and proofs involving absolute values

Is the following proof correct? Let ($\forall$ x, y $\in $$\Bbb R)$ $|x-y| \le |x| +|y|$ case#1: Suppose x and y $\ge0$. We want to show that $|x-y| \le |x| + |y|$. Since $|x|\ge x$ and $|y|\ge y$, ...
12
votes
3answers
1k views

Show that the product of two consecutive natural numbers is never a square.

I'd like to have my proof verified and if possible, to see other solutions that are interesting. Proof: Suppose $n(n+1)$ is a square. Then we write $$n(n+1) = \prod_{p} p^{c(p)}$$ where $c(p) = a(p) ...
0
votes
3answers
61 views

Sequence that contains subsequences converging to every point in the infinite set $\{1/n \} \, \forall \, n \in N$ (Abbott p 58 q2.5.3c)

has this property. Notice that there is also a subsequence converging to 0. We shall see that this is unavoidable. I acquiesce to this example, but I wasn't conscious of it until I read ...
3
votes
4answers
70 views

A more rigorous way to prove this? [duplicate]

I would like to prove the following statement $$x^n-a^n=(x-a)\sum^{n-1}_{k=0}x^ka^{n-k-1},\qquad\forall n\in\Bbb N_0$$ I can easily prove it by induction using polynomial long division or series ...
3
votes
1answer
43 views

A Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues - Proof Strategy [Lay P397 Thm 7.1.3c]

Would someone please explain the proof strategy at Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues? I brook the algebra so I'm not asking about ...
1
vote
0answers
27 views

Proving $d_E$ is a Metric on $\Bbb R^n$

This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1): $d_E(\mathbf {q,p}) = 0 \iff ...
0
votes
0answers
42 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
0
votes
3answers
48 views

Proof that $X^C \cap Y^C= \;(X \cup Y)^c$

Proof that if $X \subset S,\; Y\subset S,\;$ then $\;X^C \cap Y^C= \;(X \cup Y)^c:$ It must be shown that the two sets have the same elements, that each element of the set on the left is an element ...
2
votes
2answers
64 views

Could this linear algebra proof be done without computation?

From page 95 of Hoffman & Kunze's Linear algebra: Let $T$ be the linear operator on $\mathbb{R}^2$ defined by $T(x_1,x_2)=(-x_2,x_1)$ Prove that if $B$ is any ordered basis ...
1
vote
5answers
69 views

If one of the hypotheses holds, then one of the conclusions holds. (looking for a proof)

Using a huge truth table, I proved the theorem below. I cannot find a more elegant proof. I tried to rewrite expressions; e.g. using the distributive laws and the laws of absorption - to no avail. Is ...
0
votes
2answers
79 views

Combinatorics proof of “sum of (k choose m) with k from m up to n is equal to n+1 choose m+1”

I've already proved this statement algrebraically. I'm asked to prove it with combinatorics. So far I came up with, LHS= # ways to choose m apples from a total of m,m+1,...,n RHS= # ways to choose ...
2
votes
0answers
87 views

Choices of epsilons in proof : $(b_n) \to b$ implies $\left\{\frac{1}{b_n}\right\} \to \frac{1}{b}$ (Abbott pp 47 T2.3.3.iv) [closed]

Original became long, ergo I ask anew. The trick is to look far enough out into the sequence $(b_n)$ so that the terms are closer to b than they are to 0. Consider the particular value $e_0 = |b|/2$. ...
0
votes
2answers
58 views

Consecutive positive integers proof problem

Consider any three consecutive positive integers. Prove that the cube of the largest cannot be the sum of the cubes of the other two. Work: I tried to prove via contradiction. I made three ...
2
votes
0answers
94 views

Back-and-Forth Argument vs. “One-Way” Argument

The wikipedia article on the Back and Forth Argument claims at the end: If we iterated only step $(1)$, rather than going back and forth, then in some cases the resulting function from A to B ...
1
vote
0answers
348 views

Easier Solution? - Find plane perpendicular to another plane and through the intersection line of two planes [Stewart P803 12.5.38]

$38.$ Find an equation of the plane that's $\perp$ the plane $x + y - 2z = 1$ and passes through the line of intersection of the planes $x - z = 1$ and $y + 2z = 3$. $\bbox[3px,border:2px solid ...
1
vote
0answers
74 views

How to prove that the inverse of a persymmetric matrix is also persymmetric?

An exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...
3
votes
0answers
45 views

Using an automatic tool for checking geometric conjectures

I do a lot of research about squares, and I thought of using some automatic tool for proving / disproving some geometric conjectures. As a simple example, consider the following Square coloring ...
1
vote
2answers
134 views

How many digits do we need for a proof ??

In the question: Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx$, the value of that integral was ...
0
votes
0answers
45 views

elegant proof of Radon–Nikodym theorem

Do you know about an elegant proof of Radon–Nikodym theorem, which is not as cumbersome as the usual ones?
1
vote
1answer
84 views

How we got $z\cdot(x+y)=x\cdot y$

This is from "Test of math at 10+2 level": A vessel contains $x$ gallons of wine and another contains $y$ gallons of water. From each vessel $z$ gallons are taken out and transferred to the other. ...
2
votes
0answers
75 views

Proof for the distributivity of multiplication over addition for a Binary Field

For the standard binary field $\mathbb{F}_{2} = \{0, 1\}$. Where the operations of addition and multiplication exist, and multiplication is equivalent to logical and, and addition is equivalent to ...
3
votes
1answer
138 views

Help with a lemma of the nth root (without the binomial formula)

I have no idea of how to solve it. I would appreciate if someone gives me a hint, please. Definitions Let $\,x^{1/n}:= sup\{\, y \in \mathbb{R}: y\ge0 \text{ and } y^n\le x\, \}$ Lemma: Let ...
2
votes
3answers
878 views

Prove that in a parabola the tangent at one end of a focal chord is parallel to the normal at the other end.

Prove that in a parabola the tangent at one end of a focal chord is parallel to the normal at the other end. Now, I know prove this algebraically, and that's very easy, but I am not getting any ...
5
votes
5answers
163 views

interesting Integral , alternative solution.

Show the following relation: $$\int_{0}^{\infty} \frac{x^{29}}{(5x^2+49)^{17}} \,\mathrm dx = \frac{14!}{2\cdot 49^2 \cdot 5^{15 }\cdot 16!}.$$ I came across this intgeral on a physics forum and ...
1
vote
1answer
66 views

proving inequality $0 < x^4+2x^2-2x+1$ for $x>0$

How can I elegantly prove the inequality $0 < x^4+2x^2-2x+1$ for $x>0$. I have plotted this function in a Sage (an open source and free CAS) and I can see that there is a local min between $0$ ...
3
votes
4answers
146 views

Prove $\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A$ by the Pythagorean theorem.

How do I use the Pythagorean Theorem to prove that $$\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A?$$
0
votes
5answers
241 views

Proof of the equality of the difference of two sets iff sets are equal (direct vs. indirect)

I have a problem with the following (really) basic result: $$A\backslash B=B\backslash A \Longleftrightarrow A=B$$ More specifically, I am able to prove it only by contradiction (in particular in the ...
4
votes
3answers
261 views

Proving or Disproving the Sum in a Set

I am doing review questions for an exam and I am completely stumped on this particular question: Let A = {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32}. Prove or disprove that if I select 10 distinct ...
3
votes
1answer
371 views

Primitive recursive functions and characteristic functions. Methods of proof- examples. Illumination.

I am puzzling over a sentence in an example in a textbook, showing that a function $f$, defined by cases, is primitive recursive. Let $E$ be the set of even natural numbers. The function $f$ defined ...
2
votes
1answer
141 views

Quasi-Coherent modules over sheaves of rings (proof question)

Let $X$ be a scheme with structure sheaf $\mathcal O_X$ and let $\mathcal B$ be a quasi-coherent $\mathcal O_X$-algebra. Let $\mathcal F$ be any sheaf of $\mathcal B$-modules. In EGA I, § 9, Prop. ...
6
votes
3answers
1k views

'Every open set in $\mathbb{R}$ is the union of disjoint open intervals.' How do you prove this without indexing intervals with $\mathbb{Q}$?

In my book's exercises section I am asked to prove that every bounded, open set in $\mathbb{R}$ is the union of disjoin open intervals. Looking around the internet I have found many strategies that ...
2
votes
2answers
194 views

Is this a valid alternative proof of the sum law in limits?

So sum law of limits tell us $\lim_{n\to\infty} (a_n+b_n)=X + Y$ if $\lim_{n\to\infty} a_n = X$ and $\lim_{n\to\infty} b_n = Y$ Here is my attempt to prove it. Proof Let ...
1
vote
0answers
77 views

proof that $P^{(n)}$ are primary when $P$ is prime

I am looking for an alternate proof of the fact that in a commutative ring $R$ with a prime ideal $P$, the ideal $P^{(n)}=P^n R_P\cap R$ is primary. I understand once we localize, $P^n R_P$ is a power ...
5
votes
1answer
236 views

What are various proofs good for?

There are plenty of questions around here, which are proven to be right or wrong in various ways. I wonder, what one can learn from these differing ways of how to prove something, despite the fact ...
6
votes
1answer
3k views

Sum of irrational numbers

Well, in this question it is said that $\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$, and the owner asks for "alternative proofs" which do not use rational root theorem. I wrote an ...