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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2answers
129 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
223 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
221 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
3answers
76 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
188 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
204 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
111 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
139 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
78 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
164 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
150 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
55 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
213 views

Dividing Squares Fails to Invoke Contradiction: Two Elementary Divisibility Proofs

$x^2 \text{ is even } \iff x \text{ is even } \tag{Thm 3.12, P76}$ $\text{ Let } x, y \in \mathbb{Z}. \text{ Then } x \;\& \; y \text{ are of the same parity } \iff x + y \text{ is even.} \tag{Thm ...
5
votes
2answers
249 views

Limit superior inequalities proof

Let $a_n$ be a positive sequence. Prove that $$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$
5
votes
1answer
121 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
1answer
351 views

What is wrong with my induction proof?

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 < ...
5
votes
2answers
90 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
110 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
111 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
266 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
2answers
752 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
70 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
84 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
530 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
84 views

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

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

Proof of pythagorean theorem

Any one seen this proof before? $$\frac{d}{dx} \sin(x)^2=2\cos(x)\sin(x)$$ $$\frac{d}{dx} \cos(x)^2=-2\cos(x)\sin(x)$$ $$\frac{d}{dx} \sin(x)^2+\frac{d}{dx} \cos(x)^2=0$$ $$\sin(x)^2+\cos(x)^2=c$$ ...
4
votes
5answers
408 views

Showing $a^2 < b^2$, if $0 < a < b$

Lately, I've been stumbling with proofs of inequalities. For example: Given $0 < a < b$ Show $a^2 < b^2$ The only thing I've been able to come up with so far: $a^2 < b^2$ ...
4
votes
6answers
1k views

Prove that - for every positive $x \in \mathbb{Q}$, there exists positive $y \in \mathbb{Q}$ for which $y \lt x$

First my apologies if this question has been asked before. Exposition I'm new at learning how to prove theorems and among the given exercises from my reference material it is asked to prove the ...
4
votes
6answers
2k views

Simple proof; Show that $(4^n - 1)$ is divisible by 3 (Guided proof task)

First part of the task is just to show that $(4^n-1)$ actually is divisible by 3 for n=1,2,3,4. No problem. Second step: is to show that $(4^n -1) = (2^n-1)(2^n+1)$ No problem, just algebra. Third ...
4
votes
3answers
426 views

Proof by induction or contradiction?

I have to prove that $(4k + 3) ^2 - (4k + 3)$ is not divisible by $4$. What would be the best approach for this, proof by induction or contradiction? I've tried both and haven't got very far. Any ...
4
votes
7answers
867 views

How do I show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab?

I really need help with this question. I am required to show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab, and I'm not quite sure how. Any work/help is ...
4
votes
2answers
1k views

Prove that there exist no positive integers $m$ and $n$ for which $m^2+m+1=n^2$

The problem: Prove that there exist no positive integers $m$ and $n$ for which $m^2+m+1=n^2$. This is part of an introductory course to proofs, so at this point, the mathematical machinery should not ...
4
votes
3answers
368 views

Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]

The problem is following, prove that: $$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$ I've tried solving this problem using mathematical induction, but I ...
4
votes
4answers
824 views

Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$ [duplicate]

Prove that $$\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}.$$ This problem is in the chapter about discrete random variables, but I have no idea what to go about substituting. I can't ...
4
votes
3answers
737 views

What is the relation A = B = C called in a proof?

When writing a proof if I have the relationship $$ A = B = C $$ And I want to use that to prove $$ A = C $$ I remember there being some term for it. What is that term, and what would be an ...
4
votes
5answers
451 views

Prove $\lim _{x \to 0} \sin(\frac{1}{x}) \ne 0$

Prove $$\lim _{x \to 0} \sin\left(\frac{1}{x}\right) \ne 0.$$ I am unsure of how to prove this problem. I will ask questions if I have doubt on the proof. Thank you!
4
votes
2answers
326 views

Proof that the Irrationals are Countable

Proof: Between any two irrationals lies a rational, by the Density of the rationals in the real number system. There are only countably many rationals; therefore, there are only countably many pairs ...
4
votes
2answers
3k views

A continuous function $f$ from a closed bounded interval $[a, b]$ into $\mathbb{R}$ is uniformly continuous

I am reading (as a supplement) the book Basic Real Analysis, by Anthony Knapp. Before I proceed into reading a proof, I want to be sure that the result seems obvious. Yet, I am having trouble seeing ...
4
votes
4answers
180 views

Help in proving $ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum\limits_{k=0}^{n} \frac{1}{k!} < 3 $.

I am trying to prove this statement for all $ n \geq 1 $ using induction: $$ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum_{k=0}^{n} \frac{1}{k!} < 3. $$ I said: Base case $ n = 1 $: $$ \left( ...
4
votes
4answers
313 views

Proof of $n^2 \leq 2^n$.

I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$. My steps are: induction base case: $n=0:$ $0² \leq 2⁰$ which is okay. inductive step: $n \rightarrow n+1:$ ...
4
votes
3answers
4k views

Prove that the intersection of two equivalence relations is an equivalence relation.

I am reading this chapter of the Book of Proof, and I'm stuck at the Exercise 10 of section 11.2. It is as follows. Suppose $R$ and $S$ are two equivalence relations on a set $A$. Prove that $R ...
4
votes
2answers
213 views

Proof of $f(\{x\})=\{f(x)\}$

I thought, is this really that simple? Or am I missing a piece? This is my proof: $f(\{x\})=\{f(x)\}$. Look at $f(\{x\})$. By definition, $f(\{x\})=\{f(a)|a \in \{x\}\}$, and therefore ...
4
votes
2answers
2k 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 $1/z$
4
votes
2answers
734 views

Prove that $x^{2} \equiv -1$ (mod $p$) has no solutions if prime $p \equiv 3\pmod 4$.

Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$ Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$. I know this problem has something to do with Fermat's Little ...
4
votes
2answers
207 views

Another Epsilon-N Limit Proof Question

How to prove the limit of the following sequence using epsilon-N argument. $$a_n=\frac{3n^2+2n+1}{2n^2+n}$$ I took the limit to be $\frac{3}{2}$ and proceeded with the argument, ...
4
votes
5answers
379 views

Given a rational number $x$ and $x^2 < 2$, is there a general way to find another rational number $y$ that such that $x^2<y^2<2$?

Suppose I have a rational number $a$ and $a^2 < 2$. Can I find another rational number $B$ such that $a^2<B^2<2$? Based on the answer to this question, I thought of doing the following: ...
4
votes
2answers
151 views

Prove that: $(1+a_1)(1+a_2)…(1+a_n) \le 1 + S_n + (S_n)^2/2! + … + (S_n)^n/n!$

I am currently working on this problem from Hardy's Course of Pure Mathematics and have gotten stuck near the end. I was wondering if someone could help me determine what to go next. Question If ...
4
votes
1answer
1k 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 ...
4
votes
2answers
903 views

$f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable

Problem. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a monotone increasing function. Show that $f$ is measurable. Solution. We know that the set of discontinuites of any monotone increasing ...