Take a real number in the open interval (0,1), and write its (infinite) binary "decimal" expansion, replacing each 0 with a 2.

You have an infinite sequence of digits, e.g. 2.211122121...

Now cut that sequence to form one number with one digit, then one number with two digits, then one with three digits, etc.

2 21 112 2121 etc.

You now have a sequence of numbers that are all different, i.e. a set. And of course, no two real numbers in (0,1) give the same set.

Therefore, you have an injection from (0,1) to P(A). QED.