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
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5answers
256 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
407 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
347 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
2answers
3k 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
4answers
178 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
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
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
22k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
5
votes
4answers
1k views

Sum of cubes proof [duplicate]

Prove that for any natural number n the following equality holds: $$ (1+2+ \ldots + n)^2 = 1^3 + 2^3 + \ldots + n^3 $$ I think it has something to do with induction?
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
1k 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
709 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
228 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
4answers
272 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
439 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
131 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
226 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
227 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
2answers
169 views

What exactly is the difference between weak and strong induction?

I am having trouble seeing the difference between weak and strong induction. There are a few examples in which we can see the difference, such as reaching the $k^{th}$ rung of a ladder and ...
5
votes
5answers
106 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
137 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
120 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
80 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
193 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
2answers
96 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
301 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
926 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
210 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 ...
5
votes
2answers
115 views

How to exactly write down a proof formally (or how to bring the things I know together)?

I've always had trouble with this: I know everything I need to know but I can't connect everything I know in the way it's being wanted from me. This is what I have to do: Prove for $f : M → N$ the ...
5
votes
1answer
142 views

showing $a_n = \frac{\tan(1)}{2^1} + \frac{\tan(2)}{2^2} + \dots + \frac{\tan(n)}{2^n}$ is not Cauchy

My gut telling me that the following sequence is not Cauchy, but I don't know how to show that. $$a_n = \frac{\tan(1)}{2^1} + \frac{\tan(2)}{2^2} + \dots + \frac{\tan(n)}{2^n}$$
5
votes
1answer
80 views

Every first countable space is a moscow space.

First countable space $X$ is an example of moscow spaces. Let $U$ is an open subset of $X$ and $x\in \overline{U}$. If $\overline{U}$ is open or even a nbhood of $x$ this proposition is immediately ...
5
votes
2answers
2k views

Proof of Non-Ordering of Complex Field

Let $\mathcal F$ be a field. Suppose that there is a set $P \subset \mathcal F$ which satisfies the following properties: For each $x \in \mathcal F$, exactly one of the following statements holds: ...
5
votes
4answers
169 views

Why would the author ask if I used the Associative Law to prove + is not equiv. to *?

I just started reading An Introduction to Mathematical Analysis by H.S. Bear and problem 1 goes as follows: Problem 1: Show that + and * are necessarily different operations. That is, for any ...
5
votes
2answers
1k views

help me understand a line in an “$A^TA$ is positive, semi-definite” proof

I am looking at a proof for why $A^TA$ is positive semi-definite when $A$ is $n\times n$ and it has this line. $$ v^TAA^Tv = A^Tv \cdot A^Tv ≥ 0. $$ I understand what $v^TAA^Tv$ means and the purpose ...
5
votes
2answers
177 views

What is an instance of a mousetrap proof?

A part of the first chapter of the book The spirit and the uses of the mathematical sciences talks about the beauty of mathematics. The author quotes from a lecture of Hasse and introduces the notion ...
5
votes
1answer
57 views

Is this a valid proof of the contrapositive?

The question is the following: if $a$ and $b$ are distinct group elements, then either $a^2 \neq b^2$ or $a^3 \neq b^3$. I find this difficult to prove directly, so I formulated the contrapositive to ...
5
votes
1answer
142 views

Help to understand the proof of $ \lim \limits_{x\to 0^+} f \left(\frac{1}{x}\right)=\lim \limits_{x\to \infty}f(x)$

The following is an answer to the proof of $$ \lim \limits_{x\to 0^+}f\left( \frac{1}{x} \right)=\lim \limits_{x\to \infty}f(x)$$ If $l=\lim \limits_{x\to \infty}f(x)$, then for every ...
5
votes
2answers
95 views

Product of a family of spaces of countable tightness

I recently learned the concept of cardinal functions and some of the definitions and theorems are not clear to me. How can we prove this theorem? Finite family of compact spaces of countable ...
5
votes
1answer
272 views

$K^n \cong K^m \implies n = m$

Let $K$ be a field and let $E$ be a vector space over $K$. I want to prove that any two finite bases of $E$ are equinumerous. What I did was: Let $B = \{u_1, \cdots, u_n\}$ be a finite basis of $E$. ...
5
votes
1answer
140 views

Prove spatial velocity identity - screw theory

This question involves a proof regarding coordinate transformations of velocities of screw motions. This comes from "A Mathematical Introduction to Robotic Manipulation" (the text is available for ...
5
votes
1answer
101 views

With $N$ a constant $>0$, show $\prod_{p<x}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that if $x$ is large enough,$$\prod_{\substack{p<x \\ p \ \text{prime}}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this ...
5
votes
1answer
115 views

How to prove the inequality?

Given $0<x<1$, $0<a<b<1$, and $a+b<1$, how to prove $a^x(1-ax)<b^x(1-bx)$? I've tried using $f(x)=x^t(1-xt)$ to do some manipulations (including derivations), but failed.
5
votes
1answer
294 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 ...
5
votes
1answer
876 views

Prove that the only prime triple is 3, 5, 7 [duplicate]

Prove that the only prime triple is $3,5,7$. I tried proving using this method: Multiplication of $3$ jumps back and forth between being an even and an odd number. Thus goes from odd to odd over an ...
5
votes
2answers
847 views

Proof to sequences in real analysis

I need some verification for my proof in part a) and help to get me started on part b) a) Prove that the sequence $a_n = (2n+1)/(3n+5)$ converges to $2/3$ directly from the definition of convergence ...
5
votes
1answer
74 views

A statement about an element $a$ in semigroup S, such that $aS$ containts idempotent and $a=axa$ implies $x=xax$

I have been currently studying some characteristics of completely regular and completely simple semigroups and I have came across a lemma, which seems simple, but I'm struggling with it's proof, so I ...
5
votes
1answer
86 views

If F→G is a consequence of F, then so is ¬G→¬F. A direct proof?

Homework question (introduction to logic): "If $F \to G$ is a consequence of $\mathcal F$, then so is $\lnot G \to \lnot F$. We refer to this rule as $\to$-contrapositive. Verify this rule by giving ...
5
votes
1answer
587 views

limit superior of a sequence proof

Let $(x_{n})\in\mathbb{R}^{+}$ be bounded and let $x_{0}=\lim\sup_{n\rightarrow\infty}x_{n}$. $\forall\epsilon>0$, prove that there are infinitely many elements less than $x_{0}+\epsilon$ and ...
5
votes
1answer
70 views

need help proving an interval

I am trying to proof $$\frac {1} {ek} \le \frac {1}{k} (1 - \frac {1}{k} )^{k-1} \le \frac {1}{2k} $$ for k>=2 to prove this I first multiply by k getting $$\frac {1} {e} \le \left(1 - \frac ...
5
votes
0answers
114 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 ...