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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16
votes
7answers
3k views

how to be good at proving?

I'm starting my Discrete Math class, and I was taught proving techniques such as proof by contradiction, contrapositive proof, proof by construction, direct proof, equivalence proof etc. I know how ...
1
vote
1answer
990 views

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

I'm having difficulity solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this with induction. I started like this: The lowest natural number where the ...
144
votes
13answers
6k views

Can every proof by contradiction also be shown without contradiction?

Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantage/disadvantages of proving ...
19
votes
1answer
2k views

How would one be able to prove mathematically that $1+1 = 2$?

Is it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically?
8
votes
2answers
896 views

Prove that $\mathcal{P}(A)⊆ \mathcal{P}(B)$ if and only if $A⊆B$. [duplicate]

Here is my proof, I would appreciate it if someone could critique it for me: To prove this statement true, we must proof that the two conditional statements ("If $\mathcal{P}(A)⊆ \mathcal{P}(B)$, ...
44
votes
3answers
7k views

Proof by contradiction vs Prove the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by ...
10
votes
8answers
796 views

Book about technical and academic writing

I'm in the process of writing my Master's Thesis on automata theory. The writing must be in English which isn't my mother tongue. So the question is, given that this is my first time long (hundred ...
3
votes
3answers
401 views

How to prove that $z\cdot\text{gcd}(a,b)=\text{gcd}(za,zb)$

I need to prove that $z \cdot \text{gcd}(a,b)=\text{gcd}(za,zb)$. I tried a lot, for example, looking at set of common divisors of the two sides, but I can't conclude anything from that. Can you ...
15
votes
4answers
657 views

Advice for writing good mathematics?

It's been a (far-fetched, possibly) goal of mine to some day write a math Textbook. I've been thinking about writing this question for a while, but reading an exceedingly mediocre text on Mathematical ...
2
votes
5answers
1k views

Proof via Induction for A Summation

I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically: $$ ...
55
votes
4answers
6k views

Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?
2
votes
3answers
364 views

Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$

I am struggling to prove this map statement on sets. The statement is: Let $f:X \rightarrow Y$ be a map. i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$ ii) $\forall_{A,B \subset X}: ...
7
votes
4answers
2k views

$f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$

This is problem 6.2 from the 3rd edition of Principles of Mathematical Analysis. Problem 6.2: Suppose $f\geq 0$, f is continuous on $[a, b]$, and $\int_{a}^{b}f(x)dx = 0$. Prove that $f(x)=0$ for ...
6
votes
2answers
145 views

How to show that: $\gcd\left( {a^n-b^n \over a-b} ,a-b\right)=\gcd(n d^{n-1},a-b )$

How to show that: $$ \gcd\bigg( {a^n-b^n \over a-b} ,a-b\bigg )=\gcd(n d^{n-1},a-b ) $$ $a,b\in \mathbb Z$ where $d=\gcd(a,b)$? $\gcd$ is the greatest common divisor.
4
votes
5answers
255 views

H0w t0 prove that periodic decimal numbers are rational? $a_1…a_k(b_1b_2..b_l)={m \over n}$

Given $a_1...a_k(b_1b_2..b_l)={m \over n}$ how can I prove that periodic decimal numbers are rational? Where do I even begin?
91
votes
9answers
4k views

Why do people use “it is easy to prove”?

Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the ...
4
votes
3answers
338 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
1answer
148 views

Integral calculus proof

If $f(x)$ is continuous in $[a,b]$, prove that $ \displaystyle \lim_{n \to \infty} \dfrac{b-a}{n} \displaystyle \sum^n _{k=1} f\left( a + \dfrac{k(b-a)}{n} \right) = \displaystyle \int_a ^ b f(x)dx$ ...
4
votes
3answers
161 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 ...
4
votes
5answers
381 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: $a^2 < b^2$ ...
2
votes
1answer
895 views

Prove the integral of $f$ is positive if $f ≥ 0$, $f$ continuous at $x_0$ and $f(x_0)>0$

Prove that $\int_a^b f(x)\,dx \gt 0$ if $f \geq 0$ for all $x \in [a,b]$ and $f$ is continuous at $x_0 \in [a,b]$ and $f(x_0) \gt 0$ EDIT. Please ignore below. It is very confusing actually -.- ...
1
vote
2answers
357 views

simple binomial theorem proof

I am trying to prove this binomial statement: For $a \in \mathbb{C}$ and $k \in \mathbb{N_0}$, $\sum_{j=0}^{k} {a+j \choose j} = {a+k+1 \choose k}.$ I am stuck where and how to start. My steps ...
0
votes
2answers
1k views

Proof of Bezout's Lemma using Euclid's Algorithm backwards

I've seen it said that you can prove Bezout's Identity using Euclid's algorithm backwards, but I've searched google and cannot find such a proof anywhere. I found another proof which looks simple but ...
65
votes
14answers
7k views

Why are mathematical proofs that rely on computers controversial?

There are many theorems in mathematics that have been proved with the assistance of computers, take the famous four color theorem for example. Such proofs are often controversial among some ...
68
votes
14answers
4k views

Formal proof for $(-1) \times (-1) = 1$

Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?
17
votes
10answers
3k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisivility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose ...
10
votes
3answers
560 views

Intuition of Addition Formula for Sine and Cosine

The proof of two angles for sine function is derived using $$\sin(A+B)=\sin A\cos B+\sin B\cos A$$ and $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ for cosine function. I know how to derive both of the ...
14
votes
3answers
253 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) = ...
8
votes
3answers
342 views

Proof of Wolstenholme's theorem.?

According to the theorem : $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{p-1} =\frac{r}{q}$$ And we have to prove that $r= 0 \pmod{p^2}$. (Given $ p>3$, ...
4
votes
2answers
245 views

Completeness proof of $\ell^p$

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
4
votes
1answer
14k 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 ...
3
votes
3answers
286 views

Another way to go about proving Binet's Formula

As I showed in another question of mine, it is easy to prove that $$\tag{1}\phi^{n+1} =F_{n+1} \phi+F_{n }$$ given $F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}\text{ ; }n\geq2$. Now, extending $(1)$ ...
3
votes
6answers
312 views

Would like a proofreading of my proof

Prove that if $n\in\mathbb Z$, then $n^2$ is of the form $3q$ or $3q+1$ for some $q\in\mathbb Z$ I would like to show that 3q+2 is = 3q+1 thus $n^2$ can be of the form of 3q or 3q+1. Case one ...
0
votes
2answers
86 views

Uniqueness Proof: Related to Division Algorithms

Regarding the statement: Let $a\in \mathbb {Z}$, $b\in \mathbb {Z}$. Then there exists integers q and r such that $a = qb+r$ where $0\le r \le b$. Let $S$ $=$ $\lbrace a-qb: q\in \mathbb {Z}, a-qb ...
4
votes
2answers
112 views

formal proof from calulus

Given $f:R \to R$, $f$ is differentiable on $R$ and $\lim_{x \to \infty}(f(x)-f(-x))=0$. I need to show that there is $x_0 \in R$ such that $f'(x_0)=0$ I am trying to prove it by contradiction .... ...
4
votes
3answers
486 views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say ...
3
votes
2answers
1k 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: ...
2
votes
2answers
454 views

Infinum & Supremum: An Analysis on Relatedness

$\require{color}$ Question: I need some help in proving that $\color{green}{\text{if $k\geq 0$, then $\sup (kS) = k\sup(S)$ and $\inf (kS) = k\inf (S)$}}$, and also that $\color{red}{\text{if ...
2
votes
2answers
410 views

Prove that $A \subset B$ if and only if $A \cap B = A$

Prove that $A \subset B$ if and only if $A \cap B = A$ I'm having problems writing this proof using formal logic operators. I know the idea behind it: Since $A \subset B$, the only elements A and ...
2
votes
3answers
104 views

help for proving an equation by induction

For this equation: $$-1^3+(-3)^3+(-5)^3+\ldots+(-2n-1)^3=(-n-1)^2(-2n^2-4n-1)$$ how can I prove this by induction? When I set $n = 1$ for the base case I got: $$-1^3 + (-3)^3 + (-5)^3 + \ldots + ...
-1
votes
2answers
85 views

prove that $f:X\rightarrow Y$ is surjective if and only if $f(f^{-1}(C))=C$

I need help with proving this: $f:X\rightarrow Y$ is surjective if and only if $f(f^{-1}(C))=C$ $C\subseteq Y$ Thanks.
43
votes
13answers
3k views

What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
21
votes
1answer
804 views

How to prove convergence of polynomials in $e$ (Euler's number)

These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$ This function goes to 2. I've calculated this with sage math ...
12
votes
8answers
8k views

How do I prove that a function is well defined?

How do you in general prove that a function is well-defined? $$f:X\to Y:x\mapsto f(x)$$ I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the ...
6
votes
2answers
1k views

$\epsilon$-$\delta$ limits of functions question

I'm studying how to write epsilon-delta proofs for limits of sequences, limits of functions, continuity, and differentiability and I'm having trouble with the general methodological procedure used in ...
15
votes
4answers
591 views

Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
7
votes
2answers
7k views

Prove: If a sequence converges, then every subsequence converges to the same limit.

I need some help understanding this proof: Prove: If a sequence converges, then every subsequence converges to the same limit. Proof: Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k ...
6
votes
1answer
2k views

Inductive Proof for Vandermonde's Identity?

I am reading up on Vandermonde's Identity, and so far I have found proofs for the identity using combinatorics, sets, and other methods. However, I am trying to find a proof that utilizes mathematical ...
4
votes
4answers
360 views

Good book for learning and practising axiomatic logic

I want to learn axiomatic (Hilbert style ) logic. not just a book that says that it exist and is an good way to proof theorems. What is a good book to learn and practice this method? would like: - a ...
5
votes
5answers
1k views

How does one begin to even write a proof?

I'm in my first proof based class and I'm just having a lot of trouble writing proofs. I mean I know it's not going to come natural and it will take time, but seroiusly, how does someone begin to ...