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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6
votes
1answer
97 views

A proof by strong induction that $a_n\le3^n$ where $a_n=a_{n-1}+a_{n-2}+a_{n-3}$

I am not sure whether this is right. Can anyone verify, whether this proof is valid? Thanks! Define a sequence $\{a_n\}_{n\ge0}$ as follows: ...
6
votes
2answers
341 views

Proofs in undergraduate mathematics [closed]

What are the best introduction books to mathematical proofs in undergraduate mathematics? I know of "Proofs from the Book" by M. Aigner and G. Ziegler, but also need one that shoots to analysis kind ...
6
votes
1answer
507 views

Flaw in induction proof that the Fibonacci sequence is bounded by $(5/3)^n$

The Fibonacci sequence is defined by $a_1 = 1, a_2 = 1$ and for all $n \ge 2, a_{n+1} = a_n + a_{n-1}$. Thus the sequence begins $$1,1,2,3,5,8,13,21,...$$ Prove that for all $n \ge 1, a_n ...
6
votes
1answer
171 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".
6
votes
2answers
267 views

natural language proof assistant

I was wondering whether there has been any attempt to create a proof assistant that you write in it, in english, I mean you write your proof the usual way in TeX(maybe use a 'simpler english') then ...
6
votes
0answers
143 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
405 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
3answers
2k views

Prove that $x^3 + x^2 = 1$ has no rational solutions?

Is this enough for a proof?: $$x^3+x^2 = 1$$ I would factor and get: $x^2(x+1) = 1$ I would show that $x = \sqrt1$, which is irrational but then do I have to show more? $x+1=1$ which gives me $x=0$ ...
5
votes
8answers
995 views

Is it too much rigor to turn a set into a vector space?

I was reading some online notes on vector spaces and one authors insisted on turning a set $\mathbb{X}$ into a vector space. I thought it was quite insane but maybe I am not seeing the point. The ...
5
votes
7answers
729 views

What are some examples of subtle logical pitfalls?

Here's an example: Demonstrating that the assumption $A=B$ leads to a true statement is a vacuous truth. In order the show that $A=B$, prove that the difference $\Delta =A-B$ is zero. The subtle ...
5
votes
5answers
281 views

Proving $n^3$ is even iff $n$ is even

I am trying to prove the following statement: Prove $n^3$ is even iff n is even. Translated into symbols we have: $n^3$ is even $\iff$ $n$ is even Since it's a double implication, I ...
5
votes
5answers
1k 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 $(a-b) \mid (a^n-b^n)$. ...
5
votes
5answers
143 views

Proving $n+3 \mid 3n^3-11n+48$

I'm really stuck while I'm trying to prove this statement: $\forall n \in \mathbb{N},\quad (n+3) \mid (3n^3-11n+48)$. I couldn't even how to start.
5
votes
3answers
189 views

Prove that the real root of $x^3 + x + 1$ is irrational

Using wolframalpha.com we get that the real root of this polynomial is $-0.68233$ The only way that I have found how to prove it is using the Rational Root theorem. Using that theorem the possible ...
5
votes
3answers
227 views

In a proof that is reliant on proven theorems, does one assume the reader's familiarity with said theorems, or explicitly include their logic?

In composing a proof that is reliant on proven theorems, does one simply assume the reader's familiarity with said theorems, or does one explicitly include their logic in the new logic?
5
votes
3answers
4k views

Prove the following using induction on n (matrices)

Prove the following using induction on n: $$\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}^{n} = \begin{pmatrix} n+1 & n \\ -n & -n+1 \end{pmatrix}$$ I know that multiplication of ...
5
votes
6answers
420 views

“$n$ is even iff $n^2$ is even” and other simple statements to teach proof-writing

I am supposed to teach undergraduate students who do not major in mathematics and I would like to give them a short introduction to mathematical reasoning and to the concept of proof. I am looking for ...
5
votes
5answers
582 views

Some elementary number theory proofs.

I'm solving some problems in Apostol's Intro to Analytic Number Theory. He asks to prove $$(a,b)=1 \wedge c|a \wedge d|b\Rightarrow (c,d)=1$$ My take is $$\tag 1(a,b)=1 \Rightarrow 1=ax+by$$ $$ ...
5
votes
3answers
3k views

Proving the so-called “Well Ordering Principle”

Is there anything wrong with the following proof? Theorem. Every non-null subset $B \subset\mathbb{N}$ has a least member. Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing$, ...
5
votes
5answers
257 views

Minimum of $x^2 + 3x - 1$ on $[0,1]$ and $[-2,2]$

Consider the problem of finding the absolute minimum of the function $f : [0,1] \rightarrow \mathbb{R}$ that satisfies $f(x)=x^2 + 3x - 1$ everywhere. Suppose we suspect, by graphical methods, that ...
5
votes
2answers
4k views

Prove that such an inverse is unique

Given $z$ is a non zero complex number, we define a new complex number $z^{-1}$ , called $z$ inverse to have the property that $z\cdot z^{-1} = 1$ $z^{-1}$ is also often written as $1/z$
5
votes
1answer
2k views

are two consecutive numbers relatively prime?

I have a question. I have been given this proof: "For any $n$ in the integers where $n>2$, show there are at least $2$ elements in $U(n)$ that satisfy $x^2=1$." I have gone through and actually ...
5
votes
2answers
450 views

How do you get a paper to be peer reviewed

I have a proof that I want to undergo peer review. I unfortunately am not affiliated with any university. How should I go about getting it reviewed and either rejected or published? Thanks!
5
votes
3answers
385 views

Is “$n$ is an integer and $\frac{n}{n+1}$ is an integer” true or false?

I am working through a suggested exercise "If $n$ is an integer, $\frac{n}{n+1}$ is not an integer" - I can prove this is false, and I can prove the converse is false, and I can prove the ...
5
votes
4answers
179 views

prove $s(x+y)=s(x)s(y)$

I am asked to prove the following: Let $s(x):=\sum_{n=0}^{\infty}\binom{x}{n}$. Then $s(x+y)=s(x)s(y)$. I don't know how to start. I am thinking about $\exp(x)$ function with ...
5
votes
2answers
1k views

Showing there are no integer solution to equation $\;2^x = 4y+3$

I am stuck on this problem and I'm not sure how to approach it. Can anyone help me out with figuring how to approach the proof? My task is to: Prove that it is impossible to find integers $\,x,\, ...
5
votes
1answer
103 views

For $n\ge4$, prove that $1!+2!+\cdots+n!$ cannot be the square of a positive integer

I'm trying to prove this by induction but seem to be getting nowhere.
5
votes
5answers
2k views

Prove that given a nonnegative integer $n$, there is a unique nonnegative integer $m$ such that $(m-1)^2 ≤ n < m^2$

Prove that given a nonnegative integer $n$, there is a unique nonnegative integer $m$ such that $(m-1)^2 ≤ n < m^2$ My first guess is to use an induction proof, so I started with n = k = 0: ...
5
votes
6answers
771 views

How to make sure a proof is correct

If you come up with a proof of a mathematical proposition, how do you verify the proof is correct? Put it another way, how do you avoid a wrong proof? I guess there is no definitive answer to this. ...
5
votes
3answers
154 views

Prove ten objects can be divided into two groups that balances each other when placed on the two pans of balance. [closed]

There are 10 objects with total weight 20, each of the weight being a positive integer. Given that none of the weights exceed 10, prove ten objects can be divided into two groups that balances each ...
5
votes
2answers
4k views

Proof of $\gcd(a,b)=ax+by$

Here is my proof of $\gcd(a,b)=ax+by$ for $a, b, x, y \in \mathbb{Z}$. Am I doing something wrong? Are there easier proofs? $a,b \in \mathbb{Z}, g=\gcd(a,b)$ and suppose $g \neq ax + by$. Let $c$ be ...
5
votes
3answers
91 views

Infinite set always has a countably infinite subset

I'm trying to show that one infinite always has a countably infinite subset, but I'm confused with something on the proof. Let $S$ be one infinite set. In that case, to show it has one countably ...
5
votes
3answers
233 views

Proof of $x*0=0$ is invalid?

My professor told me today that while my logic was good and all my steps after the assumption were correct, my argument was invalid because I was assuming what I want to prove. I don't see how. ...
5
votes
3answers
286 views

Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction: How to prove one of them?

Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction How to prove one of them ? On Proofwiki there is an article proving the equivalence of the ...
5
votes
4answers
311 views

Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$

Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$. So, I can intuitively see that this is true. My proof mostly makes ...
5
votes
3answers
569 views

How to prove or statements

How do I prove statements of the following types: $A \text{ or } B \implies$ C $A \implies B \text{ or } C$ I don't know how to go about proving statements like this when they have "or". Can ...
5
votes
2answers
132 views

Is this sloppy writing for limits?

Please note that I am not asking you to compute or show me how to do this limit. I am asking how to write out a clean and formal solution that is free of any error, ambiguity, or sloppiness. Given ...
5
votes
3answers
228 views

How to show x and y are equal?

I'm working on a proof to show that f: $\mathbb{R} \to \mathbb{R}$ for an $f$ defined as $f(x) = x^3 - 6x^2 + 12x - 7$ is injective. Here is the general outline of the proof as I have it right now: ...
5
votes
2answers
231 views

Need help with very simple set theoretic proofs

I am self studying Munkres' Topology book, and I'm having a hard time writing down proofs that relate to set theory. I can see why certain arguments are true, but constructing a formal proof seems to ...
5
votes
5answers
135 views

Prove if $n^2$ is even, then $n^2$ is divisible by 4

I am working on this question Prove for every integer n if $n^2$ is even, then $n^2$ is divisible by 4. prove by contradiction Proof: Since there exists an integer $n$ such that $n^2$ is ...
5
votes
2answers
179 views

Proving that $\sin1 $(radian) is irrational without using Taylor Series Expansion.

In university last semester I was asked to prove that $\sin1$ (1 radian that is) is irrational, and ended up simply using the Taylor Series Expansion. This method provides a very quick solution, but ...
5
votes
2answers
126 views

$P(x)=x^3+ax^2+bx+c$, Proof $e^{P(x)}=\sin x$ has a solution.

Let $P(x)=x^3+ax^2+bx+c$ Proof : $e^{P(x)}=\sin x$ has a solution. I thought about it, and still cannot find where to start. Any ideas?, Thanks!
5
votes
3answers
81 views

Grade this proof of a surjective map from $\mathbb{R}^3$ to $\mathbb{R}^3$.

I just received this homework proof back in my abstract algebra class with a grade of 20%. I feel very cheated, to say the least. I present it here verbatim for your critiques. Please tell me what ...
5
votes
4answers
202 views

Use induction to prove a product of sums of squares is a sum of squares

For any natural number $n\ge 1$, given pairs $(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$ of integer numbers, there exist integer number $c$ and $d$ such that $$\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$$ My ...
5
votes
1answer
48 views

Stating the induction hypothesis

I would like to ask about the best way to state the induction hypothesis in a proof by induction. Just to use a concrete example, suppose I wanted to prove that $n!\ge 2^{n-1}$ for every positive ...
5
votes
2answers
101 views

Closed form of a Definite Integral [duplicate]

I attempted to integrate the following function from a practice problem in my Calculus textbook: $$\displaystyle \int_{0}^{\frac{\pi}{2}}{\frac{1}{1+\tan^\sqrt{2}(x)}} \ {\rm d}x$$ I failed to find ...
5
votes
1answer
360 views

For every $\epsilon >0$ , if $a+\epsilon >b$ , can we conclude $a>b$?

If $a+\epsilon > b$ for each $\epsilon >0$, can we conclude that $a>b$? Please help me to clarify the above. Thanks in advance.
5
votes
1answer
974 views

Proving that a metric space is compact

Let $H^\infty$ be the set of real sequences such that each element in each sequence has $|a_n|\leq 1$. The metric is defined as $$d(\{a_n\}, \{b_n\}) = \sum_{n=1}^\infty \frac{|a_n - b_n|}{2^n}.$$ ...
5
votes
2answers
95 views

Idea of a proof by contradiction

Is the idea of a proof by contradiction to prove that the desired conclusion is both true and false or can it be any derived statement that is true and false (not necessarily relating to the ...
5
votes
2answers
213 views

Is (the proof of) Fermat's last theorem completely, utterly, totally accepted like $3+4=7$?

If a mathematician would/does make use of Fermat's last theorem in a proof in a publication, would s/he still make use of some kind of caveat, like: "assuming Fermat's last theorem is true" or ...