It can make sense to include a short proof that any topological space satisfying certain simple conditions must be uncountable and perhaps that the reals do satisfy those conditions.  That the reals are uncountable can be proved more simply from the fact that they are linearly ordered such that between any two points there is another and they satisfy the least-upper-bound property (you certainly DON'T need to talk about decimals, even though that's one way to do it).