For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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241 views

Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
9
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533 views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
8
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294 views

Proof that $1729$ is the smallest taxicab number

For homework I have to produce the proof (algebraic or otherwise) to show that $1729$ HAS to be the smallest taxi cab number. A taxicab number means that it is the sum of two different cubes and can ...
6
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67 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
6
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373 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
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220 views

Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
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95 views

Saddle Points on Matrices

Let $n$, $m$ be positive integers. Suppose that $A$ is a $2$ x $n$ or an $m$ x $2$ matrix and that it has a saddle point. Show that among the saddle points of $A$ there exists at least one which ...
4
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71 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
4
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72 views

Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
4
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282 views

extension of Cauchy's Integral formula

This question is from Brown and Churchill's Complex Variables and Applications, 8ed., Section 52, Question 6. Let $f(s)$ denote a continuous function taken along a simple contour, $C$ enclosing a ...
4
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156 views

Easy proof of trivial fusion implies normal p-complement

Theorem: Suppose G is a finite group with Sylow p-subgroup P. Then the following are equivalent: The set K of elements of G of order relatively prime to p (the pā€²-elements) form a subgroup If A and ...
3
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33 views

An undirected graph $G$ can be decomposed into simple edge-disjoint cycles if and only if all of its vertices have even degree.

Research effort: $\rightarrow)$ I think this is relatively easy. $\leftarrow)$ Let $G = (V,E)$, let $w$ be any vertex of $G$, given that all the vertex have even degree, I'm assured that I can ...
3
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0answers
39 views

Do I need to use induction to have sufficient rigor in this proof?

I'm taking my first analysis class this summer. The professor asked us to prove that $a^{2n}-b^{2n}$ is divisible by $a+b$. After dorking around with the first couple of $n$ I was able to come up ...
3
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67 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
3
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130 views

Prove $f_\infty: A_\infty \rightarrow B_\infty$ is a bijection

Update: I was given some hints at how to approach this problem $A_\infty $ and $B_\infty$ are sets, not maps. (which is strange because there are function definitions coming into play here) The ...
3
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0answers
74 views

My first simple direct proof (very simple theorem on real numbers). Please mark/grade.

What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Let $I = [a,b]$ be a non-empty closed ...
3
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69 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
3
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94 views

Substitute Proof by Contradiction/Negation with Direct Proof or Proof by Contrapositive

When do proof by contradiction, or by negation, or direct proof, or proof by contrapositive all work equally well? When can any one be chosen as the proof form? I'm interested in a pragmatic answer ...
3
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0answers
59 views

Proofs involving Disjunctions [Velleman, Chapter 3.5]

$\Large{{1.}}$ Are proofs using strategies $P136, P143$ always easier than those using $P140$? In the former two, only one statement (either $P$ or $Q$) must be proven. In the latter, both $P$ and ...
3
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77 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
3
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79 views

A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
3
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0answers
64 views

Is there a proof reading website where users can submit their problems and upload their proof documents to be reviewed?

I'm just looking for a proof reading website that would allow a user to post her/his problem to be proved and her/his scanned documents to their claimed proof to the problem. I was just about to post ...
3
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207 views

Proving identity involving sum

I'm stuck trying to prove the following identity: $$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 ...
3
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193 views

Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.

For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$ Here is a quick one. If someone could improve this it would be great Proof By Cauchy Schwarz, $\langle x,z \rangle ...
3
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0answers
48 views

(DFM) vs (DFS) spaces, Banach scales

I have already posted this question on MO: http://mathoverflow.net/questions/126007/dfm-vs-dfs-spaces-banach-scales However, not having received any feedback, I decided to repost it, since I have the ...
3
votes
0answers
1k views

Proof of the sine rule

So I made my first attempt at a proof. I think it turned out well. Maybe not. But I was wondering if someone could take a look at it and tell me what they think. I'd be glad to hear some criticism on ...
3
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0answers
939 views

Prove that the set Aut(G) of all automorphisms of the group G with the operation of taking the composition is a group

Let $G$ be a group. Say what it means for a map $\varphi: G \rightarrow G$ to be an automorphism. Show that the set-theoretic composition $\varphi \psi = \varphi \circ \psi$ of any two ...
3
votes
0answers
206 views

Transcendence of $e$ (proof)

I'm trying to get through the proof of transcendence of $e$ (the base of the natural logarithm) already for a couple of days, but now I got seriously stuck. Proof is in most sources roughly the same. ...
3
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79 views

Solving inequalities by creating a function.

I have noticed a few answers involving inequalities proven by creating a function comprised of the terms in the inequality. I hadn't seen this before and was wanting to know more about how to use it. ...
3
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0answers
145 views

Do mathmatician ever prove that a theorem could not generalize into a much general theorem? Is there a historic mile stone example?

Do mathmatician ever prove that a theorem could not generalize into a much general theorem? Is there a historic mile-stone example refer to the above question?
2
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0answers
14 views

Set with relative complement forms partition

Prove that if $S$ is a set and $ \emptyset \subsetneq A \subsetneq S $ then $\Pi = \{A , S-A \}$ is a partition of $S$. Proposed Solution: Since $ A \subsetneq S$ , we have $S - A \neq ...
2
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0answers
56 views

Simple Proofs in ZFC Set Theory

So I'll keep this real short and simple. In this document on page 23 there is a list of axioms. On page 24 there is a list of theorems that come from said axioms. I can prove them all except 3 and 5. ...
2
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0answers
34 views

Velleman's How to prove it. Partial order proof.

Theorem: Suppose that $R$ is a partial order on $A$, $B_1 āŠ† A$, $B_2 āŠ† A$, $x_1$ is the least upper bound of $B_1$, and $x_2$ is the least upper bound of $B_2$. Prove that if $B_1 āŠ† B_2$ then ...
2
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0answers
61 views

Proof regarding derivatives and Mean Value Theorem.

Original question: $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Show that for any $c \in (a,b)$ that is not a point of maximum or minimum for $f'$, there exist $x_1, x_2 \in (a,b)$ ...
2
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0answers
111 views

Can you share your experience in hand waving and other informal communication regarding mathematical proofs?

I intend to write a paper that will address among other issues the informal communication between mathematicians. My point of origin is the view that every proof can be represented by a sequence of ...
2
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0answers
73 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
2
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0answers
101 views

Finding minimum graph of $k$ disjoints component

Sorry for the english, I tried to make it the most clear possible. Be $G$ a connected graph not directed, I have to find an algorithm that given $n$ the quantity of vertex and $0<k<n$ disjoint ...
2
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0answers
29 views

On ambiguity in statements expressed in natural language, where the statements use an indefinite article, e.g. “a”.

Please consider the following example statements and judge the meaning of the article "a". Example: A house is a building. Example: A house is being built next to our house. In example 1, "a" is ...
2
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0answers
48 views

My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
2
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0answers
43 views

Help needed in understanding a proof

Claim: Let $M$ be a $R$-module ($R$ is an integral domain) and $p \in R$ be a prime. Suppose there exists non-empty finite subsets $B$ and $C$ of $M \backslash\{0\}$ such that $M= \bigoplus_{m \in ...
2
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0answers
86 views

Bretschneider-Brahmagupta-Heron Proof

Derive Bretschneider's formula, Brahmagupta's formula and Heron's formula in one memorable elegant proof. I ask this question merely to see the creativity of the MSE community when it comes to ...
2
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26 views

Selecting a unique pair satisfying a condition $\varphi$ with an ordering

Given a finite structure $\mathfrak{A}$ with Universe $|A| < \infty$ and signature $\tau$. We say a pair $(a,a') \in A$ satisfies a $\tau$-formular $\varphi$ iff $$ \mathfrak{A} \models ...
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65 views

Set of limit points of S is closed in a metric space X

A point $x \in X$ is a limit point of a subset S of X, if every ball $B(x;\varepsilon)$ contains infinitely many points of S. Show that x is a limit point of S iff there is a sequence {$x_{j}$} ...
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0answers
77 views

If $a\lt 0$ and $b\lt 0,$ then $ab\gt0$.

$\quad$The following assertion is somewhat less obvious: If $a\lt 0$ and $b\lt 0,$ then $ab\gt0$. The only difficulty presented by the proof is unraveling of definitions. The symbol $a\lt 0$ means, ...
2
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0answers
143 views

Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.

I would like to know if my proof below is correct. Problem Let $p$ be a prime. Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles. Show by ...
2
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0answers
99 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
2
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0answers
18 views

Prove that $(-\infty, b)$ for any $b \in \mathbb{R}$ has cardinality $c$

Prove that $(-\infty, b)$ for any $b \in \mathbb{R}$ has cardinality $c$ Proof: Define function $f: (-\infty, b) \rightarrow (-b, \infty)$ by $f(x) = -x$. Note that $f$ is one-to-one and onto because ...
2
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0answers
40 views

Prove that $f: \mathbb{N} \rightarrow \mathbb{N}-\left \{ 1 \right \}$ given by $f(x) = x+1$ is $1$-$1$ and onto

$f: \mathbb{N} \rightarrow \mathbb{N}-\{1\}$ given by $f(x) = x+1$ is $1$-$1$ and onto. Proof: ($1$-$1$) Suppose $f(x_{1}) = f(x_{2})$ for $x_{1}, x_{2} \in \mathbb{N}$. Then $x_{1} + 1 = x_{2} + ...
2
votes
0answers
72 views

Prove $\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 + b^2}\right) \ge 9$

If $a,b,c \in \mathbb{R^+}$,then prove that the following inequality holds: $$\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 ...
2
votes
0answers
54 views

On the proof that the inverse value set of a regular value is a submanifold

I have a doubt on the proof of the following, well-known theorem: Let $f:M^m\rightarrow N^n$ ($m\geq n$) be a $C$ map, $r\geq 1$.If $y\in f(M)$ is a regular value, then $f^{-1}(y)$ is a $C$ ...