<p>Let me clear something up here, all jokes aside.</p>
<p>0.999… is <em>not</em> a limit, it’s a well-defined representation of a certain real number. The question that we have is whether or not the representation 0.999… applies to the same real number as the representation 1.000…</p>
<p>Rigorously speaking, I don’t like the proofs like:
1/3 = 0.333…
3/3 = 3(1/3) = 3(0.333…) = 0.999…
That sort of presupposes a little too much about the algebra of real numbers to make the proof too useful…</p>
<p>Also, proofs involving limits of summations really don’t say anything about the actual number 0.999…, just about limits of sequences of rational numbers approaching the number. Mathematical induction typically cannot be trusted at infinity, and I don’t see why summations need be any more special.</p>
<p>For my money, the most useful proof that 0.999… represents the same real number as 1.000… relies on the definition of the real numbers… perhaps the one using Dedekind cuts, for instance. Proofs based on simple properties of the real numbers are pretty straightforward.</p>