6 days, 17 hours ago

Proofs

Hello all! Following my post about the importance of definitions in one's work as a student of mathematics, this post discusses the notion of proof and how one can begin to write their own proofs.

At the core of being a mathematician, you often need to provide an argument for why a particular pattern/trend, which you believe you entirely understand, holds under some set of conditions that you have determined. This is where the importance of proof appears, because the mathematical community is the body that checks if such a pattern, if it holds, should then be made to be part of general knowledge and mentioned in mathematical journals.

So, what is a proof? Here's a dictionary definition of proof:


A sequence of steps, statements, or demonstrations that leads to a valid conclusion.               - Dictionary.com


The key question then is how does one create this sequence of steps that the above definition speaks of? Well, as a first step you need to distinguish between all of the assumptions/givens of a statement and its conclusions. We use the statement of the Bolzano-Weierstrass theorem as an example:

Every bounded sequence of real numbers contains a convergent subsequence.

The givens/assumptions of this statement are:

  • ' Every... ' is equivalent to ' For any given ... '
  • Bounded sequence of real numbers

And the conclusion consists of:

  • ' ... contains a ... ' means to say that ' ... there exists a...' or ' ... one can find a... '
  • convergent subsequence

The terms in bold are where your knowledge in definitions would be of great assistance to not only understanding what the overall statement means exactly but also to get a hint at how to create the proof. So, in beginning to prove a statement, it may be a general good start to extract the important details as I have done here.

Another way to derive statements, such as the Bolzano-Weierstrass theorem, is to use combinations of previous results in a legitimate way to arrive at the desired conclusion. These results may sometimes be called propositions or lemmata. These are initial results that are often easier proved on their own before using them together to prove a theorem. The tricky part, however, is figuring which simpler results you need to prove in order to make the path towards proving the theorem much easier. My best advice would be to do some research on results of a relevant field that have already been established.

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