[x42bn6]

I'd give you a full formal definition about Dedekind cuts, the Archimedian property and a full rigorous lesson on limits but I'll refrain from it. No proof is sadly complete without them (what is a limit? Is the limit of an infinite sum equal to the sum of the limit of the underlying sequence?).

All intents and purposes aside, see 0.999... - Wikipedia, the free encyclopedia... which explains things better than anyone could possibly do on this forum. Except probably MacMan.*