For questions about the formulation of a proof, not about the mathematics behind it.

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10
votes
0answers
935 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 ...
9
votes
0answers
52 views

What to keep in mind when attempting proof of basic properties of divisibility/what techniques are useful/what's the intuition for showing them?

So I am currently trying to prove some basic divsiibility relations, as follows. If $a \mid b$ and $a \mid c$, then $a \mid (b + c)$. If $a \mid b$ and $s \in \mathbb{Z}$, then $a \mid sb$. ...
6
votes
0answers
143 views

How do people pick $\delta$ so fast in $\epsilon-\delta$ proofs

For example, in a proof that shows $f(x) = \sqrt(x)$ is uniformly continuous on the positive real line, the proof goes like: Let $\epsilon > 0$ be given, and $\delta = \epsilon^2$.... Or to ...
6
votes
0answers
130 views

Why so many 'multi-part' definitions, as opposed to 'unified' ones?

Many definitions consist of multiple parts: an equivalence relation is symmetric AND reflexive AND transitive; a topology is closed over finite intersections AND over arbitrary unions; etc. However, ...
6
votes
0answers
607 views

In a math paper, what is a remark?

I sometimes see paragraphs labeled 'Remark.' However, papers that include remarks also include unlabeled explanatory paragraphs (i.e. all the other writing in the article) that seem to be remarks. ...
6
votes
0answers
163 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all the positive integers $m$ such that both the ratios $$ \frac{2(5^m+5)}{3^m+1}, \frac{9^m+1}{5^m+5}$$ are integers. Attempt to a solution: If the ratios are both integers, than their ...
6
votes
0answers
422 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, ...
5
votes
0answers
87 views

Proving not equicontinuity in $\Bbb R$ but equicontinuity in any other closed subset of $\Bbb R$

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. Prove that $F$ is not equicontinuous on $\Bbb R$ but equicontinuous on $[−a, a]$ for any $a ...
4
votes
0answers
70 views

How to prove $\forall x,y\in\mathbb{R}: x^2+y^2 = (x+y)^2 \Leftrightarrow x=0\lor y=0?$

The question I really have is the structure and I am not sure to use pack-unpack or not. Here is my try: Let $x,y\in\mathbb{R}$ Assume $x^2+y^2 = (x+y)^2$ Then $x^2+y^2 = x^2+2xy+y^2$ #by ...
4
votes
0answers
61 views

Proof of inequality involving binomial coefficients

Proving things is not my forte. Stumbled upon following identity: $$\binom{n+k+1}{k}>\sum_{i=1}^k \binom{\alpha_i}{i}$$ For $\alpha_i<\alpha_j $ for $i<j$ $0\leq \alpha_i \leq n+k$ Also $...
4
votes
0answers
111 views

How to Prove It, Exercise 6.5.9

Suppose $R$ is a relation on $A$ and $S$ is the transitive closure of $R$. If $(a, b) \in S$, then there is some positive integer $n$ such that $(a, b) \in R^n$, and therefore by the well-ordering ...
4
votes
0answers
63 views

Proving $f$ cannot be onto

If $f$ maps finite sets $A$ to $B$ and $n(A) < n(B)$, prove that $f$ cannot be onto. Proof by contradiction: If $f: A→B$ and $n(A) < n(B)$, $f$ is onto. Since, by definition of a function, $...
4
votes
0answers
185 views

$\zeta(2)=\frac{\pi^2}{6}$ proof improvement.

Recently in one of my calculus exercise I have made out a (quite novel to me) proof for $\zeta(2)=\frac{\pi^2}{6}$ via the famous infinite product below: $$\sin(x)=x\prod_{i=1}^{\infty}(1-\frac{x^2}{i^...
4
votes
0answers
56 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
4
votes
0answers
56 views

Prove this congruence

Let $p$ be a prime of the form $4k+3$ and $m$ an even positive integer less than $p-1$. Prove that $$1^m+2^m+\cdots+\left(\frac{p-1}{2}\right)^m \equiv \left(\frac{p+1}{4}\right)+\left(\frac{p+1}{4}\...
4
votes
0answers
47 views

Another proof question for real analysis

Let $a, b, c \in \mathbb{R}$. Prove if $a + b = a$ then $b = 0$. Suppose that $a + b = a$. Then $a + b - a = a - a = 0 = b$ by the inverses law for addition. By the Identity law for addition it ...
4
votes
0answers
465 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 $\mathscr{C}^1$ on $(c,d)$. Then ...
4
votes
0answers
166 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 ...
4
votes
0answers
94 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
votes
0answers
718 views

The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.

I am trying to prove the Lebesgue Criterion for Riemann Integrability without using the concept of oscillation. The Lebesgue Criterion for Riemann Integrability states that if $ f: [a,b] \to \mathbb{...
4
votes
0answers
281 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. ...
4
votes
0answers
272 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 p′-elements) form a subgroup If A and ...
3
votes
0answers
40 views

Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...
3
votes
0answers
33 views

Recreational math dealing with twin primes

This is kinda recreational math with a goal in mind of progressing further toward a proof of the twin prime conjecture. Consider this: We start with a random prime: $109$ $3*109=327$ $327 \equiv$...
3
votes
0answers
35 views

Expected number of kitchens sold (Exercise 3.11.38 from Grimmett and Stirzaker)

A telephone sales company attempts repeatedly to sell new kitchens to each of the $N$ families in a village. Family $i$ agrees to buy a new kitchen after it has been solicited $K_i$ times, where the $...
3
votes
0answers
45 views

Complete Graphs as Unions of Paths

Show that for $n \geq 2$ the complete graph $K_n$ is the union of paths of distinct lengths. I have been stuck on this problem for the past couple of days now and would really like to see a solution/...
3
votes
0answers
63 views

A different way to prove homeomorphism between rectangles and discs under standard $\Bbb R^2$ topology

I was reading about the classical problem to prove homeomorphism between subspaces of $\Bbb R^2$ a rectangle of the kind $R=\{(x,y):|x|\le a \land |y|\le b\}$ and some disc $D=\{(x,y): x^2 + y^2 \le r\...
3
votes
0answers
37 views

Proving a Particular Coefficient of a Power Series Equals $0$

Suppose I have a particular function $$F(x,z) = \sum_{n=0}^\infty{A_n(x)\frac{z^n}{n!}}$$ and suppose, through the use of a particular computer algebra system, that the particular polynomial ...
3
votes
0answers
60 views

Is there any mistake in my proof?

My little brother started fiddling around with his calculator, and noticed something curious: $$ \Large \sqrt{a\cdot\sqrt{a\cdot\sqrt{a\cdot\sqrt{a \cdot \ldots}}}} = a $$ So I went ahead and wrote a ...
3
votes
0answers
40 views

Leibniz Notation for the Derivative of a Function

I am writing a professionally-written proof, and I have come across a bit of an issue regarding how to write the derivative of a function $H(t)$ with respect to $t$. Is $\frac{dH}{dt}$ an acceptable ...
3
votes
0answers
25 views

Looking for references for learning the words and sentences used in proofs

I'm familiar with textbooks on logic, proof techniques, and sets. But I have yet to encounter a textbook that dives into the language used w/ definitions and sentence structure used in proofs, for ...
3
votes
0answers
112 views

If $f$ is continuous and injective on an interval, then it is strictly monotonic- what's wrong with this proof?

I've tried proving the following theorem: Let $a,b\in \mathbb{R}$ and $f:[a,b]\rightarrow \mathbb{R}$. If $f$ is continuous and injective, then it is also strictly monotonic. This is my attempt at a ...
3
votes
0answers
51 views

Exercise on limits: is this proof correct?

Prove that $$\lim_{x\rightarrow0}x^\alpha \ln{x}=0$$ for every $\alpha>0$ This is my attempt: Let $g(x)=x^x$ $f(x)=x^\alpha$ $h(x)=g\circ f(x)=(x^\alpha)^{(x^\alpha)}$ $k(x)=x^{(x^\alpha)}$ ...
3
votes
0answers
166 views

Geometric proof for Sophomore's dream

Is there a "visual proof" for sophomore's dream? $$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty n^{-n}.$$ In the wikipedia article there are two algebraic proofs, but the integral and the summation has ...
3
votes
0answers
42 views

Much more topology…

These are very similar to my last two questions. I provide them with my thoughts so far: $(1)$ Let $S$ be the collection of all straight lines in the plane which are parallel to the x-axis. If $S$ is ...
3
votes
0answers
28 views

Help on proving an observation related to pythagorean triangles.

Working for a while on Pythagorean triangles I observed that if $n$ has the prime factorization $p_1^{r_1}p_2^{r^2}...p_i^{r_i}$ where $r_k$'s not all even Then we have: $\frac{(r_1+1)(r_2+1)...(...
3
votes
0answers
78 views

Help with a trigonometric proof, please?

Hexagon $ABCDEF$ is inscribed in the circle of radius $R$ . $AB=CD=EF=R$. Points $I$, $J$, $K$ are the midpoints of segments $\overline{BC}$, $\overline{DE}$, $\overline{FA}$ respectively. Then prove ...
3
votes
0answers
86 views

Using the Intermediate Value Theorem to prove a statement about an equation true

I want to prove this statement true by using the IVF: For any real number $b > 2$, the equation $2^x = bx$ has a solution. Here are some questions I need help with answering: Define a function $...
3
votes
0answers
93 views

Is this proof of a mathematical olympiad problem correct?

I'm quite sure about the exactness of my proof, but I'd like to hear (constructive) criticism about my writing. This is the problem: Every non-negative integer is coloured white or red, so that: 1) ...
3
votes
0answers
82 views

Problem on symmetric polynomials

The following problem is from "Analysis I" by Amann/Escher. Exercise: There are obvious operations of $S_m$ on $\mathbb{N}^m$ and on $R[X_1,\dots,X_m]$. A polynomial $p\in R[X_1,\dots,X_m]$ is called ...
3
votes
0answers
103 views

Real analysis question: Suprema and Infima

Let $S$ be an ordered set with the $L.U.B$ Property, $S \supset B \neq \varnothing$, $B$ is bounded below. Write $L = \{ l : l \; \text{is a lower bound of } \; B \} $. Then, it follows that $\...
3
votes
0answers
315 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
votes
0answers
97 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 $x_1, x_2 \in (a,b)$ ...
3
votes
0answers
170 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 $...
3
votes
0answers
128 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
votes
0answers
128 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
votes
0answers
308 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
votes
0answers
197 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
votes
0answers
87 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
votes
0answers
538 views

Proof verification for limit point, lim sup and lim inf

Hi everyone I'd like to know if the next argument is sound. I'd appreciate any suggestion. Thanks :) Proposition: Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers, let $L^+$ be the limit ...