7
votes
2answers
144 views

Soft question: Unconventional proofs

I'm not sure if I understood it correctly, but one of my professors told us that one theorem was proved this way: A mathematician assumed the truth of the Riemann hypothesis and was able to prove a ...
46
votes
17answers
2k views

Rigour in mathematics

Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious. Can you give me examples of conjectures/theories that seemed true but through rigorous ...
1
vote
1answer
112 views

A question on mean value inequality

It is known that mean value inequality is very useful. It is: For any $0 \le a_i (i=1,2,\dots,n)$, $$ a_1 a_2\dots a_n\le (\frac{a_1+a_2+\dots + a_n}{n})^n \tag1 $$ My question is: how many ...
4
votes
0answers
165 views

Good examples of proofs in mathematics exemplary of creative reasoning [closed]

Just what the title says. I'm not looking for any proofs that require specialized knowledge past the very fundamentals of real analysis. I'm looking for proofs for important results (don't have to be ...
10
votes
8answers
236 views

Examples of “transfer via bijection”

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then ...
20
votes
5answers
1k views

Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis

I am studying for an exam and I have been studying my butt off during the winter break for it. During the course of my study I have written down quite a number of tricks, which in my opinion were ...
16
votes
5answers
1k views

“Too simple to be true”

As indicates the title, this question is about "proofs" of true statements which are short and/or look elegant but are wrong. I mean example like Cayley-Hamilton's theorem, which states that for a ...
5
votes
3answers
95 views

Simple Characterizations of Mathematical Structures

By no means trivial, a simple characterization of a mathematical structure is a simply-stated one-liner in the following sense: Some general structure is (surprisingly and substantially) more ...
2
votes
3answers
151 views

Proofs for Undergraduates

I am teaching a course in proof technique to undergraduate students. One of the things they can do for their project is read an involved proof and explain it, for example Gödel's Incompleteness ...
2
votes
0answers
190 views

Examples of proofs by induction with respect to relations that are not strict total orders.

I have read this Wikipedia article and found it fascinating. I came across it after I tried to prove a certain statement with a method resembling induction in the set of natural numbers but ordered by ...
43
votes
13answers
3k views

Pseudo Proofs that are intuitively reasonable

What are nice "proofs" of true facts that are not really rigorous but give the right answer and still make sense on some level? Personally, I consider them to be guilty pleasures. Here are examples ...
17
votes
6answers
2k views

What are some common proof strategies in mathematics?

I want to start out by saying that I am new at proof based mathematics. I am used to seeing patterns and using them to solve similar problems. However, I have found this is not a very good way to ...
47
votes
7answers
6k views

Lesser-known integration tricks

I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now ...
11
votes
3answers
709 views

A good book for learning mathematical trickery

I've seen several question here on what book to read to learn writing and reading proofs. This question is not about that. I've been doing that for a while, and I'm quite comfortable with proofs. I am ...