It's probably more convenient to show an injection in each direction, from which you can conclude that the cardinalities are equal.
For P(N)->R, map a subset S of N to the real number less than 1 given in decimal notation by putting a 1 in the nth place after the decimal if n is in S, and a 0 otherwise. Of course, ternary or some other base would work, but not binary--you don't want to mess with non-unique representations of the same real number.
For R->P(N), choose a real number x in (0,1), and represent it in binary. Then do the same thing as above, but in reverse. Every rational number with a power-of-2 denominator has two such representations; choose the terminating one. A bijection between R and (0,1) is all you need to finish (actually, just an injection into (0,1) will do).
To do LaTeX: