Suddenly had this idea as I was doing my GP maths tutorial. My sis and I used to argue about it.

1/9 = 0.111...
2/9 = 0.222... and so on until,

8/9 = 0.888
Therefore, can we conclude that 0.999... = 9/9 = 1?

Then, I suddenly thought of it in terms of GP!
0.999... is actually a GP in disguise. You can see it as: 0.9 + 0.09 + 0.009 + 0.0009 + ...
which is equals to 9/10 + 9/100 + 9/1000 + 9/10000 + ... and goes on infinitely. (read: sum to infinity).

You can also see that the common ratio here is simply 0.1 or 1/10, which is < 1, which means we can use the simplified form of finding the sum of a convergent series!

First term: 9/10 or 0.9
Common ratio: 1/10 or 0.1
So sum to infinty is basically a/(1-r) = 0.9/(1-0.1) = 0.9/0.9 = 1

WOOT yeah so I proved it :D

Koped this from someone's status: