Introduction to Proofs

Proofs are, indeed, a difficult thing in mathematics. But mathematics is at its core just proofs, and less importantly calculations.

To prove requires a certain ingenuity to construct the right object or synthesize ideas in the right order and then show that that proves what you think it does.

For example, prove that the interval \([0,1]\) and \([0,1)\) have the same number of elements. This means you can find a one-to-one function (a bijection) between them. Try it, see if you can figure it out.

I wrestled with this problem for a long time. I tried drawing graphs but they all had a curve, meaning they fail the horizontal line test.

It turns out, that you have to construct a clever object:

\(f:[0,1]\to [0,1)\) by the following:

For all fractions \(\frac{1}{n}, n\in\mathbb{N}\):

\(f:\frac{1}{n}\mapsto \frac{1}{n+1}\)

for all else:

\(f:x\mapsto x\)

You may have heard of this, it’s commonly called Gilbert’s Hotel. In order to map between infinities where one of them has “extra” elements. You shift \(1\) to \(\frac{1}{2}\), \(\frac{1}{2}\) to \(\frac{1}{3}\), and so on. This is indeed a bijection.

Being exposed to problems and solutions like this increases your ability to solve similar problems. Especially in fields like analysis, we require increasingly more and more machinery and ingenuity to solve difficult problems. To prove something new requires even more of the same.

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