Some of you may have seen the YouTube video of Vi Hart in which she presents a couple of proofs that 0.999... = 1. In case you haven't, here's the link:
9.999... reasons that .999... = 1

While watching this video, I came up with another proof that feels very natural. It uses geometric series and in particular the following equality which holds iff |q| < 1

to get rid of the infinite sum. Without further ado, let's get to the proof.

So that's it, the argument whether or not 0.999... = 1 has been settled in yet another way.