Show that the set of all subsets of an infinite enumerable set is not enumerable I know this problem involves using Cantor's theorem, but I'm not sure how to show that there are more subsets of an infinite enumerable set than there are positive integers. It seems like a lot of these problems are really the same problem, but they require some unique and creative thought to get them just right. Any idea how I can solve these more quickly? What train of thought do you go though when working with this specific example?

Given an enumerable set $S=\{x_1,x_2,\ldots\}$, and an infinite sequence of 0's and 1's, $\{b_1,b_2,\ldots\}$, we can associate a subset of $S$ consisting of those $x_i$ for which $b_i=1$. This is a bijection between colection of subsets of $S$ and infinite binary sequences. These binary sequences can be through of as base 2 representation of all real numbers between  0 and 1, hence it is uncountable.  
