One equals almost one

I came accross this one recently, and thought it was interesting. Feel free to discuss.

We wish to prove that . (That is to say, 0.99999…, a repeating decimal.)

Why would we want to prove something so seemingly rediculous? Well, we take for granted such things as approximation and rounding, and are more than happy to take limits as they approach infinity, so it would be interesting to prove that using one is just as good as using 0.99999…, if the situation allows.

• Let
• And
• Then:

• Thus,
• But since as well, this goes to say ! Which is what we wished to prove.

Some interesting side notes… the logic in the proof plays out differently for other common repeating decimals. Take for instance. Run through the same procedure and you will come out with which reduces to . So there's no trick involved in the proof, but it's also worth noting that is not representable as a rational number either.