Since there was some discussion of infinity in the whole .999... shebang, I thought I'd post a quick proof related to infinite sets. The result is well-known: the set of rational numbers and the set of natural (counting) numbers have the same cardinality (that is, they are the same size). This is normally shown with Cantor's diagonalization proof.
This result is usually considered counter-intuitive because there are infinitely many rational numbers just between 0 and 1 alone. Yet, the set of rationals is indeed the same size as the set of natruals. All I wanted to do in this post was give you my favorite proof of this. I know there are many. I know you'll be tempted to post your favorite in the comments (feel free). But for some reason, this one is so clever, it's by far my favorite. I learned it from a professor in college.
I will prove that the set of (positive) rational numbers can be associated one-to-one with a subset of the natural numbers. Since it is easy to associate the natural numbers with a subset of the rationals (the reciprocal function will do), the demonstration below completes the proof.
By definition, each positive rational number can be expressed in a unique manner as p/q, where p and q are natural numbers with no common factors. Consider the string of symbols composed of (digits of p)/(digits of q), with a slash between the strings of digits. Now read that string (which is unique for every rational) as an integer in base 11, with the slash being the symbol for 10. There's your integer associated with that rational number. QED.