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

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6
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4answers
371 views

Is my proof that $(A^n)^{-1} = (A^{-1})^n$ correct?

I am still learning Linear Algebra at it's basic levels, and I encountered a theorem about invertible matrices that stated that: If $A$ is an invertible matrix, then for $n=0,1,2,3,..$. $A^n$ is ...
6
votes
5answers
934 views

Prove that there is no number that divides both n and n+1

Statement There is no number $x > 1$ that divides both $n$ and $n+1$. Proof (my attempt) Indirect proof: \begin{align} x\mathbin{\vert} n & \implies n = xt_1 \\ x\mathbin{\vert}(n+1) ...
6
votes
3answers
84 views

Construct a sequence of measureable sets $E_1\supseteq E_2 \supseteq E_3 \supseteq \cdots$ such that $\mu(E_n)=\infty$ for each $n$ but …

Construct a sequence of measureable sets $E_1\supseteq E_2 \supseteq E_3 \supseteq \cdots$ such that $\mu(E_n)=\infty$ for each $n$ but $$\mu\left(\bigcap_{n=1}^\infty E_n\right)=0$$ Claim: Let \...
6
votes
2answers
6k 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 ...
6
votes
5answers
315 views

Show that $\gcd(a,b)=\operatorname{lcm}(a,b)$ if and only if $a=b$.

I know how to prove $a=b$ only if $\gcd(a,b)=\operatorname{lcm}(a,b)$, but I don't know how to prove the "if part". Can anyone help me?
6
votes
4answers
6k views

Proving the so-called “Well Ordering Principle”

Is there anything wrong with the following proof? Theorem. Every non-empty subset $B \subset\mathbb{N}$ has a least member. Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing$, ...
6
votes
5answers
2k views

What character can replace word “let” in proofs?

For example, suppose I have a line of a proof introducing new “variable” $x$: $$\textrm{Let}\:\:x\in f(y)$$ I am looking for ways to express the word “let” in this context and ...
6
votes
2answers
7k 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 $1/z$
6
votes
4answers
512 views

Proving a theorem, what is meant by sufficiency and necessity?

I am looking at the proof of a theorem and the proof begins by saying ...is the proof of the sufficiency part of this theorem so we just need to establish the necessity of the condition. What ...
6
votes
1answer
565 views

Disproving existence statements

I am trying to get some practice on disproving existence statements and I was really stuck on this one: "There exists an example of three distinct positive integers different from $a,2a,3a$ for some ...
6
votes
7answers
805 views

prove: if n is even, then n+1 is not even

This proof seems so simple that it's hard (if that makes any sense.) based on the definition, n is even iff there exists k such that n = 2k. What I really want to say is (big picture) By definition,...
6
votes
7answers
3k views

Looking for Proofs Of Basic Properties Of Real Numbers

I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there's one thing that I need to clarify and that is the properties of real ...
6
votes
5answers
62 views

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing.

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing. I do not have use derivatives, so I decided to apply the definition of being a strictly increasing ...
6
votes
5answers
207 views

how to prove $\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$

i am asked to prove this statement: $$\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$$ Now after several attempts, i am lost not knowing where and how to start. if I use induction, i am stuck on ...
6
votes
2answers
5k 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 ...
6
votes
3answers
268 views

How would I prove $|x + y| \le |x| + |y|$?

How would I write a detailed structured proof for: for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$ I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ ...
6
votes
3answers
240 views

Let $f:A\rightarrow B$. Prove that if $ X\subseteq A$, and $f$ is one to one, then $f(A)-f(X) \subseteq f(A-X)$.

Let $f:A\rightarrow B$. Prove that if $ X\subseteq A$, and $f$ is one to one, then $f(A)-f(X) \subseteq f(A-X)$. Could anyone please guide me through this problem? I got stuck and don't know if what ...
6
votes
3answers
114 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
2answers
4k views

Prove the inequality $n! \geq 2^n$ by induction

I'm having difficulty solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this by induction. I started like this: The lowest natural number where the ...
6
votes
5answers
129 views

What is the most used method for proving continuity for simple functions such as $f(x) = x^{1/3}$

In analysis we talked about a very general definition of continuity: $f:A \to B$ is continuous if $U \subset B$ is open, $f^{-1}(U) = V \subset A $ is open Quite elegant Another definition is if $(...
6
votes
2answers
117 views

Prove $(3x^2+3) \geq (x+1)^2+1$

$(3x^2+3) \geq (x+1)^2+1$ I tried using a direct proof but I think I got stumped along the way. $3x^2+3 \geq x^2+2x+2$ $2x^2+1 \geq 2x$ $2(x^2) +1 \geq 2x$ $x^2 + (1/2) \geq x$ How can I make ...
6
votes
4answers
585 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 ...
6
votes
4answers
334 views

How to show that a limit cannot be another number?

Let: $$ G(x) = \left\{ \begin{array} {cc} x \sin \frac{1}{x} , & x\neq 0 \\ 0, & x=0 \end{array} \right. $$ I can understand that the function is continuous at $x=0$ because: For $\epsilon&...
6
votes
1answer
623 views

Proving the Poincare Lemma for $1$ forms on $\mathbb{R}^2$

I am trying to prove the Poincare Lemma for $1$ forms on $\mathbb{R^2}$. So I said that if I doing this, I start of with $$\omega = f_1(x_1,x_2) dx_1 + f_2(x_1,x_2)dx_2.$$ First thing I want to ...
6
votes
2answers
984 views

Can you prove why Popsicle Stick Multiplication works?

This is a unique way of multiplying numbers by using sticks. Let's call it "Popsicle Stick Multiplication". Or maybe "Linear Algebra" quite literally. Take a look at both images that I've drawn ...
6
votes
4answers
168 views

prove $\lceil{x}\rceil=-\lfloor-x\rfloor$

i am trying to prove that $\lceil{x}\rceil=-\lfloor-x\rfloor$, but having difficulties to prove. the definitions are: $\lceil{x}\rceil:=m-1<x\leq m$ and $\lfloor{x}\rfloor:=n\leq x<n+1$. how ...
6
votes
3answers
126 views

How to structure long proofs

How do you structure proofs that are longer than say half a page? I have already encountered a variety of styles (in my short math life), some of which I list below and I just hoped to hear some wise ...
6
votes
4answers
198 views

Prove that if $A \mathop \triangle B \subseteq A$ then $B\subseteq A$

I'm having trouble with the logic in this proof and was wondering if anyone could point me in the right direction (if I'm wrong)? Prove that if $A\mathop\triangle B\subseteq A$ then $B\subseteq A$. (...
6
votes
1answer
489 views

Supposed proof of dirichlets theorem on primes

I think theirs somthing wrong with this proof as it was not hard to create, if someone could find a mistake I would greatly appreiciate it: Define a function $[k\equiv b \bmod a]$, to be equal to ...
6
votes
2answers
3k 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 ...
6
votes
2answers
66 views

prove/disprove that if $f\circ g$ injective and g is surjective, then f is injective

Question would be: prove/disprove that if $f\circ g$ injective and g is surjective, then f is injective. after thinking, I came to the conclusion that it's a proof. tried to prove it but it looks not ...
6
votes
4answers
273 views

Any commutative associative operation can be extended to a function on nonempty finite sets

This is a fact we use very frequently in general mathematics when we write such notations as $1+2+3+4$: since we know that $+$ is commutative and associative, we can just "drop the parentheses" and ...
6
votes
1answer
42 views

Show that if $x^2 y=2x+y$, then if $y \neq 0$ then $x \neq 0$

Prove if $x^2 y=2x+y$, then if $y \neq 0$ then $x \neq 0$. Obviously, $x,y \in \mathbb R$. I know this is rather simple. It is more about the process than this example. Is it logically correct to do ...
6
votes
2answers
186 views

How can I prove whether a $9\times 9$ square can be filled with L-shaped pieces in a completely “regular” way?

There are a great many ways to fill a $9\times 9$ square with L-shaped pieces. One of them is below. Now, note that there are eleven $2\times 3$ rectangles that are formed, as well as a larger L ...
6
votes
2answers
293 views

Sipser Pumping Lemma Clarification

In a Theory of Computation book I am using, the explanation of Pumping Lemma is not bad, but some parts of it are not clear to me. Here is the Definition of Pumping Lemma: If A is a regular language,...
6
votes
2answers
242 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 ...
6
votes
1answer
523 views

Rudin's 'Principle of Mathematical Analysis' Exercise 3.14

Since I'm studying real analysis using this book by myself, I'm not sure whether or not my method to prove convergence of sequence is right. I'm working on the above question's (d), and my solution ...
6
votes
3answers
8k views

Set Distributive Property Proof

Prove the distributive property for sets: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ I'm not good with proofs but my understanding is that I have to prove 2 things: (1) $A \cup (B \...
6
votes
1answer
69 views

Prove that P(A) ∪ P(B) ⊆ P(A ∪ B).

I have a presentation on this Monday. I thought it was pretty straight forward but my professor wrote "You need to show why x is in P(AUB), not just state that it is." I thought that I had. Here's ...
6
votes
3answers
332 views

Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$.

Question : Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$. Hint: consider decimal expansions ...
6
votes
3answers
330 views

Proving {$b_n$}$_{n=1}^\infty$ converges given {$a_n$}$_{n=1}^\infty$ and {$a_n b_n$}$_{n=1}^\infty$

Suppose {$a_n$}$_{n=1}^\infty$ and {$b_n$}$_{n=1}^\infty$ are sequences such that {$a_n$}$_{n=1}^\infty$ coverges to A$\neq$0 and {$a_n b_n$}$_{n=1}^\infty$ converges. Prove that {$b_n$}$_{n=1}^\...
6
votes
1answer
129 views

The author of my book simplifies his solutions to an extent that I am uncomfortable with, so are my solutions to homework over doing it?

This question can be summarized as: How explicit does one need to be when writing proofs? To what extent can one implicity write a proof safely? The first chapter of our text in elementary discrete ...
6
votes
2answers
112 views

Is it good to use mean value theorem in $\epsilon-\delta$ continuity proofs?

I wanted to prove $f(x) = \cos(x)$ is continuous using $\epsilon-\delta$ proof Couple of posts on MSE appealed to MVT to resolve this problem. Namely: $\exists c \in [x,x_o]$ s.t. $|\cos(x)-\cos(...
6
votes
1answer
126 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: $$a_0=1,\qquad,a_1=3,\qquad,a_2=9,\qquad,a_n=a_{n-...
6
votes
2answers
369 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
2answers
8k views

Prove that $\sup(A+B) = \sup(A) + \sup(B)$ and why does $\sup(A+B)$ exist?

We want to show that $\sup(A)+\sup(B)$ is the least upper bound of the set $A + B$. First, we need to show that $\sup(A) + \sup(B)$ is an upper bound for the set $A + B$. Indeed, if $z\in A + B$, then ...
6
votes
1answer
612 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
2answers
326 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
4answers
391 views

Prove A is an open set if and only if $A \cap Bd(A) = \emptyset $

Prove A is an open set if and only if $A \cap Bd(A) = \emptyset $ Here is my start: Suppose A is an open set. We know $X-A$ is closed. Need to show $A \cap Bd(A) = \emptyset$ Let $ x \in A$. ...
6
votes
1answer
175 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".