Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

1,005 questions
5k views

Proof writing: how to write a clear induction proof?

What is an effective way to write induction proofs? Essentially, are there any good examples or templates of induction proofs that may be helpful (for beginners, non-English-native students, etc.)? ...
12k views

How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
15k 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 ...
23k 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?
14k views

Proof: How many digits does a number have? $\lfloor \log_{10} n \rfloor +1$

I read recently that you can find the number of digits in a number through the formula $\lfloor \log_{10} n \rfloor +1$ What's the logic behind this rather what's the proof?
57k views

30k 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 ...
13k views

Differentiability implies continuity - A question about the proof

I have a question, to aid my understanding, about the proof that differentiabiility implies continutity. Differentiability Definition When we say a function is differentiable at $x_0$, we mean that ...
15k 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?
7k 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?
9k 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 ...
14k views

Logarithm proof problem: $a^{\log_b c} = c^{\log_b a}$

I have been hit with a homework problem that I just have no idea how to approach. Any help from you all is very much appreciated. Here is the problem Prove the equation: $a^{\log_b c} = c^{\log_b a}$ ...
24k 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 proving ...
11k 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 ...
4k views

Show that if $g \circ f$ is injective, then so is $f$.

The Problem: Let $X, Y, Z$ be sets and $f: X \to Y, g:Y \to Z$ be functions. (a) Show that if $g \circ f$ is injective, then so is $f$. (b) If $g \circ f$ is surjective, must $g$ be surjective? ...
13k 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 $\mathcal{P}(A)⊆ \mathcal{P}(B)$, ...
489 views

Proving the Apollonian circle formula

Proposition: Let: $c,d\in\mathbb{C} \space : c\neq d$ Then: $\forall k \in \mathbb{R}^{+}\setminus\{1\} \forall z \in \mathbb{C}: |z-c|=k|z-d|$ represents Appolonius circle. What was already done: ...
143k views

Limit of $(1+ x/n)^n$ when $n$ tends to infinity [duplicate]

Does anyone know the exact proof of this limit result? $$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x$$
1k views

Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.

Problem: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$. My work: So I think I have to do a proof by induction and I just wanted some help editing my proof. My attempt:...
570 views

3k 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 ...
293 views

Stuck with proof for $\forall A\forall B(\mathcal{P}(A)\cup\mathcal{P}(B)=\mathcal{P}(A\cup B)\rightarrow A\subseteq B \vee B\subseteq A)$

I came to point where I suppose for case 1 that $A\subseteq B$ and conclusion is trivial. For case 2 I suppose that $A\not\subseteq B$ and try to prove $B\subseteq A$, but that gets me nowhere. Any ...
11k views

Proving the sum of the first $n$ natural numbers by induction [duplicate]

I am currently studying proving by induction but I am faced with a problem. I need to solve by induction the following question. $$1+2+3+\ldots+n=\frac{1}{2}n(n+1)$$ for all $n > 1$. Any ...
642 views

Singular value decomposition proof

I need help in the following question. I'm not sure how to even begin to answer this. What is a possible proof for the following question? If $A$ is an $m \times n$ matrix and $b$ is an $m$-vector, ...
1k 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?
4k views

A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$

While browsing on Integral and Series, I found a strange integral posted by @Sangchul Lee. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind ...
172k views

What is the integral of 0?

I am trying to convince my friend that the integral of $0$ is $C$, where $C$ is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is $0$.
4k views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
17k 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 ...
6k views

Proving an infinite number of primes of the form 6n+1

The proofs given on other sites weren't that clear and used different methods that I have yet to learn. Prove that there are an infinite number of primes of the form 6n+1. The hint that was given ...
2k views

Complex numbers - Exponential numbers - Proof

Let $z$ be a complex number, and let $n$ be a positive integer such that $z^n = (z + 1)^n = 1$. Prove that $n$ is divisible by 6. For this problem I am stumped...how should I begin? Also there's a ...
993 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 $\color{red}{\text{if$k<...
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$. [duplicate]
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$. Also... $a,b$ and $n$ are natural numbers. I feel I should begin with EEA to multiply out the gcd's, but I don't know where to go from ...