Tagged Questions

4
votes
0answers
58 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 ...
2
votes
1answer
94 views

Providing a sketch for a proof before proceeding through the actual proof. [closed]

Question is pretty straightforward. My mathematics is sloppy, and I recognize my inaptitude in that my proofs are more or less too intuitive. My diagnosis dictates the fact that I attack a problem ...
-2
votes
1answer
90 views

Proving By Subsets [closed]

I am currently trying to learn about conducting proofs by using subsets. In my textbook, there is an example on this very matter; however, the seem to do something that is in contradiction with what ...
10
votes
7answers
194 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 ...
3
votes
3answers
147 views

How to do diagram chasing effectively?

I am trying to teach myself some homological algebra, and the book I am using is Aluffi's wonderful Algebra: Chapter 0, which introduces homology at the end of chapter 3. I have spent a lot of time ...
5
votes
3answers
192 views

In a proof that is reliant on proven theorems, does one assume the reader's familiarity with said theorems, or explicitly include their logic?

In composing a proof that is reliant on proven theorems, does one simply assume the reader's familiarity with said theorems, or does one explicitly include their logic in the new logic?
1
vote
2answers
144 views

What is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?

What is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory? I was thinking that textbooks make the proofs over complicate. If ...
8
votes
9answers
445 views

Example of a conjecture/theorem which required an entirely new idea to prove

When Andrew Wiles proved Fermat's Last Theorem, he built upon ideas from elliptic curves which already existed. Is there an example of a conjecture/theorem which was proved using an unexpected ...
2
votes
2answers
93 views

What makes a sufficient proof?

This question is related to the question posted here. Would a shorter proof to those in the answers, such as: Take the subsequence $\{a_m\}$ of $\{a_n\}$ where $m > 0$. By induction on $m$ ...
18
votes
6answers
650 views

How to learn from proofs?

Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I've met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or ...
25
votes
7answers
925 views

Must we use induction to prove a statement for all integers

This question is prompted by a remark from Bill Dubuque in his answer to this question on proving a particular sum without using mathematical induction. From Bill's answer: A proof that a ...
2
votes
0answers
102 views

How likely is it that some questions only have “proofs by cases” as answers? [closed]

The four color theorem's only widely known proof is of course Appel and Haken's computer-assisted one. How likely is it that this the only proof, and might there be some way to prove that this is so? ...
3
votes
2answers
521 views

Deepest theorems with simplest proofs [closed]

Which are the deepest theorems with the most elementary proofs? I give two examples: i) Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function ii) Proof that the halting problem is ...
14
votes
5answers
1k 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 ...
7
votes
2answers
466 views

Infinite number of mistakes in a proof

Writing my Bachelors Thesis has opened my eyes to what seems to be a horrible paradox. I am turning my thesis in this Friday, and have been proof reading for weeks now. Every time I print my thesis, ...
4
votes
1answer
135 views

What is a statement?

This perhaps is not a math problem. I donot know if it fits in here. But it confuses me a lot these two days. I have wiki it. But it does not work for me. I will give some easy "proofs" to explain ...
6
votes
2answers
488 views

Mathematical Telescoping

Bill Dubuque has answered several questions by indicating that some form of "telescoping" is taking place. See this post and the links provided by Bill for more information. I have never heard of ...
3
votes
2answers
307 views

Is it true that for all proofs of the statement, $\forall x \exists y : R(x, y)$, then we can say $y = y(x)$? (Example given)

Is it true that for all proofs $\forall x \exists y : R(x, y)$, then $y = y(x)$? A while back I remember reading a book on functional programming that was leading into some questions about what ...
33
votes
5answers
2k views

Getting better at proofs

So, I don't like proofs. To me building a proof feels like constructing a steel trap out of arguments to make true what you're trying to assert. Oftentimes the proof in the book is something that I ...