Questions tagged [proof-explanation]

This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

10133 questions
16
votes
4 answers

Intuitive understanding of the uniqueness of the Fundamental Theorem of Arithmetic.

Basically I am trying to understand why Fundamental Theorem of Arithmetic (FTA) exists, i.e why a natural number cannot be factored primely in two or more different ways. There are two proofs given on the wikipedia page for the uniqueness, Via…
user103816
  • 3,745
  • 6
  • 34
  • 58
15
votes
3 answers

Proof that the continuous image of a compact set is compact

Let $X\subset \mathbb R^{n}$ be a compact set, and $f :\mathbb R^{n}\to \mathbb R $ a continuous function. Then, $F(X)$ is a compact set. I know that this question may be a duplicate, but the problem is that I have to prove this using real analysis…
Lessa121
  • 1,158
  • 2
  • 8
  • 18
15
votes
6 answers

Fallacious proof involving trigonometry

Which step in the following incorrect proof is fallacious? Is it something with the use of indefinite integrals or with the domain and range of trignometric functions? I encountered this fallacious proof…
15
votes
3 answers

Proving $\operatorname{Var}(X) = E[X^2] - (E[X])^2$

I want to understand something about the derivation of $\text{Var}(X) = E[X^2] - (E[X])^2$ Variance is defined as the expected squared difference between a random variable and the mean (expected value): $\text{Var}(X) = E[(X -…
15
votes
2 answers

Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism

I need to prove two trivial results but I don't know how to work with restricted function and its inverse Consider the topological spaces $(X, \mathcal{T}), (Y, \mathcal{J})$ Claim 1: Let $f:X \to Y$ be continuous function, $A \subset X$ equipped…
15
votes
4 answers

Prove there are no prime numbers in the sequence $a_n=10017,100117,1001117,10011117, \dots$

Define a sequence as $a_n=10017,100117,1001117,10011117$. (The $nth$ term has $n$ ones after the two zeroes.) I conjecture that there are no prime numbers in the sequence. I used wolfram to find the first few factorisations: $10017=3^3 \cdot 7 \cdot…
zz20s
  • 6,362
  • 7
  • 23
  • 40
14
votes
3 answers

Finding all possible proofs

I'm now working on a geometry problem I'll have to explain in front of my class this week (I'm in the $10^{th}$ grade). I've found so far some proofs, which might, nevertheless, be a bit complicated for my classmates (since they've barely worked…
Dr. Mathva
  • 8,438
  • 2
  • 12
  • 45
14
votes
4 answers

When learning mathematics should one prove everything one learns?

I sat through a real analysis class around a year ago and in about two days we partially covered the construction of the real numbers as equivalence classes of Cauchy sequences. Through the teacher didn't do it, it took me about $9$ hours to read…
14
votes
4 answers

If X is second-countable, then X is Lindelöf.

Munkres in his book states that: Theorem 30.3 Suppose that $X$ has countable basis, then every open covering of $X$ contains a countable subcollection covering $X$. $\textbf{Proof.}$ Let ${B_n}$ be a countable basis and $\mathcal{A}$ an open cover…
George
  • 3,405
  • 1
  • 11
  • 27
14
votes
4 answers

How do people pick $\delta$ so fast in $\epsilon$-$\delta$ proofs

For example, in a proof that shows $f(x) = \sqrt x$ is uniformly continuous on the positive real line, the proof goes like: Let $\epsilon > 0$ be given, and $\delta = \epsilon^2$.... Or to show that every Lipschitz continuous function is uniformly…
13
votes
6 answers

Why is infinity multiplied by zero considered zero here?

I watched an online video lecture by some professor and she was solving a convergence problem of the power series $$\sum_{n=1}^\infty n!x^n,$$ i.e., she was finding the values of $x$ for which this power series is convergent. She did the ratio test…
13
votes
7 answers

Proof of Lemma: Every integer can be written as a product of primes

I'm new to number theory. This might be kind of a silly question, so I'm sorry if it is. I encountered the classic lemma about every nonzero integer being the product of primes in Ireland and Rosen's textbook A Classical Introduction to Modern…
13
votes
4 answers

Rudin's proof on the Analytic Incompleteness of Rationals

In Rudin's classical "Principles of Mathematical Analysis," he gave a proof like this: Claim: Let $A= \{p\in \mathbb{Q} | p>0, p^2 <2\}$. Then A contains no largest number. Proof: Given any $p\in A$. Let $q = p-\frac{p^2 -2}{p+2}$. Later Rudin…
13
votes
9 answers

Proof of infinitely many prime numbers

Here's the proof from the book I'm reading that proves there are infinitely many primes: We want to show that it is not the case that there only finitely many primes. Suppose there are finitely many primes. We shall show that this assumption leads…
13
votes
2 answers

When do $n$ ants in cyclic pursuit with constant velocity converge?

I'm reading a paper (Ants, Crickets and Frogs in Cyclic Pursuit) and trying to understand one of the simpler results. The following is a paraphrasing of the parts I'm using, but check the paper if more context is needed: Consider $n$ moving ants…