I had an undergraduate pose an interesting question to me. "Why doesn't Cantor's Diagonal Argument apply to the rationals?" http://www.proofwiki.org/wiki/Real_Numbers_are_Uncountable/Cantor%27s_Diagonal_Argument Now obviously it doesn't since the rationals are countable. But what breaks the argument? It seems obvious that the resulting "diagonal number" won't be rational since the decimal expansion of rationals either terminate or repeat. But actually proving that this "diagonal number" can't be rational seems like it would be difficult. What do you guys think?