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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5
votes
5answers
631 views

Prove that $(a-b) \mid (a^n-b^n)$

I'm trying to prove by induction that for all $a,b \in \mathbb{Z}$ and $n \in \mathbb{N}$, that $(a-b) \mid (a^n-b^n)$. The base case was trivial, so I started by assuming that $(a-b) \mid (a^n-b^n)$. ...
3
votes
0answers
110 views

Proving $\left\| \frac{\vec{v}}{\|\vec{v}\|}\right\| =1$, $\vec{v}\ne \vec{0}$ [duplicate]

I've been trying to prove that $\left\Vert\dfrac{\vec{v}}{\Vert\vec{v}\Vert}\right\Vert=1, \quad \vec{v}\ne \vec{0}$. This is my attempt: \begin{align} \vec{v}&\in \mathbb{R}^n, \quad ...
0
votes
1answer
79 views

Finding the ratio of two sides of a triangle with known angles

I wondered what the ratios between the sides of a triangle is, when the angles are known. So basically: $\triangle ABC$ has angles $\alpha, \beta \text{ and } \gamma$. Find $\frac{\lvert AB ...
3
votes
1answer
160 views

Divisibility by a prime number

I have been struggling with this question. It would be great if somebody can really help me out with this question: Prove that for any Prime number P > 5, there exists a K such that 1111....11 ...
0
votes
1answer
62 views

Help on Proof involving integrals

Good night. I'm starting to learn proofs and I'm facing the following question. Given the linear function $f(x)$, prove that $[\int_{0}^{1} f(x)\,dx]^2 < \int_{0}^{1}[f(x)] ^2dx$ As $f(x)$ is a ...
1
vote
3answers
563 views

Prove that $d^n(x^n)/dx^n = n!$ by induction

I need to prove that $d^n(x^n)/dx^n = n!$ by induction. Any help?
4
votes
3answers
112 views

Proof that $n \in \mathbb{N}$ by combinatorial analogue?

(Disclaimer: I'm a high school student, and my highest knowledge of mathematics is some elementary calculus. This may not be the correct terminology.) A while ago, I saw the following problem: prove, ...
11
votes
6answers
2k views

Proof that there are infinitely many positive rational numbers smaller than any given positive rational number.

I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$." This is my attempt of proving it:- Assume that ...
2
votes
1answer
245 views

Geometric proof that if n is a non-perfect square, then √n is irrational.

I know there is a geometric proof of the irrationality of √2. I thought maybe this one could be generalized for √n when n is a non-perfect square, but I could not find something like that anywhere. ...
3
votes
2answers
366 views

Archimedean Property - The use of the property in basic real anaysis proofs

I've been looking for something like this in the previous answers on the topic, but I didn't enounter anything similar, so here there is my problem. First of all, here there is the definition of ...
0
votes
1answer
61 views

Use of Without loss of generality (WLOG)

I encountered following usage of WLOG Consider problem of minimizing $E:= (y-x)^2 + \lambda |x|^\tau$ and $\tau \in (1,2)$ and optimizing variable is $x$, we can assume without loss of generality that ...
0
votes
3answers
399 views

$C⊆A$ and $D⊆B$ and A and B are disjoint, then C and D are disjoint.

Let A,B,C and D be sets. How to prove: $C\subseteq A$ and $D\subseteq B$ and $A$ and $B$ are disjoint, then $C$ and $D$ are disjoint. Could anyone please explain to me how to approach this ...
14
votes
3answers
262 views

Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$

For any odd positive integer $k\geq1$, the sum $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$. I used induction principle for the solution but cannot prove it. I took $P(k) = ...
13
votes
7answers
1k views

Prove that the additive inverse of an odd integer is an odd integer

This is a homework problem, but I don't want the answer, just a little guidance: Prove that the additive inverse of an odd integer is an odd integer. When approaching a problem like this, how ...
1
vote
1answer
592 views

$f$ is constant if derivative equals zero

Suppose $f'(x)=0$ for all $x\in (a,b)$. Prove that $f$ is constant on $(a,b)$. This seems painfully obvious, but I can't prove it rigorously. $f'(x)=0$ for all $x\in (a,b)$ means that for any ...
0
votes
1answer
29 views

Is $X$ has a strong rank 1-diagonal?

Definition 1: A space $X$ has a strong rank 1-diagonal \cite{5} if there exists a sequence $\{\mathcal U_n: n\in \omega\}$ of open covers of $X$ such that for each $x\in X$, $\{x\}=\bigcap ...
1
vote
1answer
74 views

Can you prove this three-way linear map composition?

OK, this was an example that my prof gave when talking about surjective, injective and bijective functions. I also am curious if I am approaching this the right way. (Everyone here has been a really ...
2
votes
1answer
117 views

Where is the fallacy in this coupling argument of two Bernoulli variables?

With respect to the scenario introduced in Prove the monotonicity of the expectation of a binary random variable function, let us now suppose that the function: $$\begin{align*} f(\mathcal{J}) = ...
0
votes
1answer
211 views

Proving a lemma about the union of set, linear independence, spans

OK, here's another lemma I'm being asked to prove and I am trying to see if I am in the right ballpark. Let $L$ = {$y, y_1, y_2, y_3,..., y_m$} be a linearly independent subset of $V$, a vector ...
1
vote
1answer
95 views

Aquaintance problem in discrete math. induction proof.

I'm supposed to prove this by induction. I already proved it by contradiction, but I am lost on how to set it up for induction. Prove that if at least two people are at a party, at least two of ...
4
votes
4answers
101 views

What is the definition of a labeled function?

I always see that people label their functions by giving an index. Specifically I have this example: $Theorem$: There is a unique binary operation $+:\mathbb{N}\times\mathbb{N}$ that satisfies the ...
0
votes
2answers
360 views

Let $R$ be an equivalence relation on a set $A$, $a,b \in A$. Prove $[a] = [b]$ iff $aRb$.

Hello I need help with the proof strategy for this problem. Let $R$ be an equivalence relation on a set $A$ and let $a,b \in A$. Prove that $[a] = [b]$ if and only if $aRb$.
0
votes
0answers
83 views

Hamiltonian cycle problem: how to prove NP-completeness?

How to prove that finding a Hamiltonian cycle in a graph is an NP-complete problem? Should I try to reduce the travelling salesman problem (TSP) to this one (Hamiltonian cycle)?
4
votes
2answers
177 views

Prove that one of the following sets is a subspace and the other isn't?

OK, here goes another. Prove that $ W_1 = ${$(a_1, a_2, \ldots, a_n) \in F^n : a_1 + a_2 + \cdots + a_n = 0$} is a subspace of $F^n$ but $ W_2 = ${$(a_1, a_2, \ldots, a_n) \in F^n : a_1 + a_2 + ...
0
votes
1answer
110 views

Is this proof complete? Case of a metric space

I am looking to prove whether or not the following statement is true: Let $M$ be a set and $D_1$ and $D_2$ metrics on M such that they form a metric space. Then $(M, \min(D_1 , D_2))$ forms a ...
0
votes
3answers
122 views

Proof regarding factorials.

Suppose $a$ and $k$ are positive integers, then how would you prove(not intuitively) that: $a!k! \leq (ak)!$ Although it is apparent that the inequality is correct, but how can I show this ...
0
votes
4answers
83 views

$a^{\log_b(c)} = c^{\log_b(a)}$ [duplicate]

I'm not sure how to start. My questions is how do you prove: $$a^{\log_b(c)} = c^{\log_b(a)}$$ where $a,b,c > 0$.
1
vote
3answers
98 views

$3^{3n+1} < 2^{5n+6} $ for all non-negative integers $n$. Is my induction solution correct?

Show using mathematical induction that $3^{3n+1} < 2^{5n+6} $ for all non-negative integers $n$. I'm not sure whether what I did at the last is valid? Basis step: for all non-negative integers ...
1
vote
3answers
59 views

What do you use for your basis step when its domain is all integers?

Example: *For all integers $ n , 4( n ^2 + n + 1) – 3 n ^2$ is a perfect square what should i use? negative infinity? I know you can use a direct proof but what if theres an induction question with ...
1
vote
1answer
245 views

$G$ is a tree $\iff G$ contains no cycles, but if you add one edge you create exactly one cycle - is my proof correct?

$G$ is a tree $\iff G$ contains no cycles, but if you add one edge you create exactly one cycle Could anyone please be so kind to check my proof? That would be very much appreciated. Thank you in ...
2
votes
0answers
84 views

Checking an inductive proof on a combinatorial product

Consider the following product, for $n, k, i \in \Bbb Z_+, k \geq 2$: $$ {\prod_{\ell = 1}^i {n + k - \ell \choose k } \over \prod_{\ell = 1}^{i-1} { k + \ell \choose k}} \tag{$*$} $$ It has been my ...
0
votes
1answer
141 views

Mathematican proof or physicist proof of a theorem?

I am writing a document, where I have proved a so-called theorem coming from a physicist paper. It is not a mathematical theorem in the sense that have only done the computations and did not check ...
3
votes
2answers
199 views

Proving there are no integer solutions for $3x^2=9+y^3$

Prove there are no $x,y\in\mathbb{Z}$ such that $3x^2=9+y^3$. Initial proof Let us assume there are $x,y\in\mathbb{Z}$ that satisfy the equation, which can be rewritten as $$3(x^2-3)=y^3.$$ So, ...
3
votes
1answer
684 views

Equivalence relation - Proof question

Prove that the relation, two finite sets are equivalent if there is a one-to-one correspondence between them, is an equivalence relation on the collection $S$ of all finite sets. I'm sure I know the ...
1
vote
4answers
579 views

Standard logic notation in mathematics

My profesor is always complaining that my proofs are very long and difficult to read because I never use notation, meaning I say everything in words. Tired of that I decided to study logic by myself ...
2
votes
3answers
94 views

Need help with Cantor-Bernstein-Schroeder Proof at ProofWiki

This concerns Proof 6 of the CBST theorem at ProofWiki. I am stuck on the line beginning "Similarly, let $g' = $" The 2nd equality on this line is not immediately obvious to me. How do you prove ...
2
votes
4answers
676 views

Prove that $\{x_1=1,\,x_{n+1}=\frac {x_n}2+\frac 1{x_n} \}$ converges when $n \to \infty$

I want to prove that the sequence defined by $\{x_1=1,\,x_{n+1}=\frac {x_n}2+\frac 1{x_n} \}$ has a limit. By evaluating the sequence I notice that the sequence is strictly monotonically decreasing ...
9
votes
3answers
137 views

Please review my question and solution. Thanks in advance.

How many values of x are there such that there exists positive integer solutions for S, such that $S=\sqrt{x(x+p)}$ where $x$ is an integer and $p$ is a prime number $>2$ This is a problem I made ...
2
votes
3answers
62 views

Prove Satisfiability of Property by Set

What is a proof strategy for proving that some property is satisfied by a particular set of numbers. For example, what would be an approach for proving that the archimedean property is satisfied by ...
4
votes
6answers
97 views

Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?

How can I show that $$ \sum_{i=1}^{n-1}i=\binom{n}{2}? $$ This is what I have tried, but I do not know if it is correct: Proof. Let $n=2$. Then, $$ \begin{align} \sum_{i=1}^{1}i&=1\text{, ...
0
votes
1answer
64 views

Could someone help me to improve the proof writing?

I will prove the following claim. I'm not a native English speaker. Could someone help me to improve the writing? A regular pseudocompact Moore space is ccc and first countable. Prove: I will ...
1
vote
1answer
490 views

How much room is there for original mathematics research?

Should more universities and colleges offer degrees in mathematical research? I am in the process of incorporating a non-profit college and I am considering offering a degree in mathematical ...
2
votes
5answers
152 views

For $a, b \in \mathbb Z,\;$ if $\;a^2(b^2-2b)$ is odd, then a and b are odd. Proof check.

Suppose $a,b$ are integers, if $a^2(b^2-2b)$ is odd, then a and b are odd, is my solution the best way? PS: I know this is easy but do i need to expand the final answer? because im practicing for ...
0
votes
1answer
36 views

Definition by Recursion and a Question about Induction

I have some questions to ask. Suppose I want to define some sequence of propositional formulas $\{\varphi_{j}\}_{j\in\mathbb{N}}$. First, I define it this way. Fix an enumeration ...
2
votes
1answer
100 views

A question about quantifiers

I'm trying to prove this theorem: Let $F$ and $G$ be functions. Then $F=G$ if and only if $\operatorname{Dom}(F)=\operatorname{Dom}(G)$ and $\forall X (X\in \operatorname{Dom}(F)\rightarrow ...
2
votes
1answer
58 views

Linear Algebra dependent Eigenvectors Proof

Problem statement: Let $n \ge 2 $ be an integer. Suppose that A is an $n \times n$ matrix and that $\lambda_1$, $\lambda_2$ are eigenvalues of A with corresponding eigenvectors $v_1$, $v_2$ ...
6
votes
3answers
104 views

How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction

I've done the base step, but how do I prove it is true for $n+1$ without using a fallacy? $$4(n!)>2^{n+2}\quad \text{for } n\geq 4$$ Please help.
6
votes
1answer
154 views

British maths style guide

For British maths style, is this punctuation OK? so if $x=-3$, then $\left|x\right|=3$, and if $x=7$, then $\left|x\right|=7$, etc with commas before "then" and "and".
1
vote
2answers
273 views

Proving a biconditional statement with an or

I want to prove a theorem in geometry of the form $p \iff q \vee r$. My plan is to prove: $q \implies p$ as well as $r \implies p$ $p \text{ and } \lnot q \implies r$ Can I get someone to verify ...
0
votes
3answers
85 views

$\log\left(\binom{n}{x} \pi^x (1-\pi)^{n-x}\right)=x\log \pi + (n-x)\log(1-\pi)\;\;?$

$$\log\left(\binom{n}{x} \pi^x (1-\pi)^{n-x}\right)=x \log \pi + (n-x)\log(1-\pi)$$ this is what i have. i dont understand how $\binom{n}{x}$ disappears, but the rest is fine. I tried this, but it ...