I was studying Cantor's contributions to set theory, and I came upon two results:
#1: It can be shown that the power set of the naturals has a higher cardinality than the naturals by attempting to enumerate all the subsets of and being able to create a new subset previously not listed, thus contradicting the fact that there was originally a bjiection between and .
#2: It can also been shown that the reals are "more numerous" than the naturals by using Cantor's Diagonalization argument.
Thus, it is known that both and have a higher cardinality than the naturals .
But how can we show that and have the same cardinality as each other, though? I tried constructing a bijection (because I know that one must exist); I tried to use Cantor's style; nothing really worked.
I am so stuck on this proof that I do not even know where to start. Someone give me a hint or a starting point, please?