How to prove "The power set of a countable set must be uncountable"?
This is Cantor's diagonalization lemma:

Let $S$ be a countable set and suppose that there is a bijective function $\phi: S \to 2^S$.  Let $U = \{x \in S \mid x \notin \phi(x)\}$.  (Notice that our only assumption is the existence of the bijective function.)

Since $U \in 2^S$, and $\phi$ is bijective, there must be a $u \in S$ s.t. $\phi(u) = U$.

Now I ask you:  Is $u \in U$?

If you answer that question, you have the answer: $\phi$ cannot exist!  Hence $2^S$ cannot be countable.  You should also show that there is an injective function $\psi: S \to 2^S$, but this is trivial: $\psi(x) = \{x\}$, so $|S| \leq |2^S|$.

To complete the argument you must also show that there cannot be a bijective function from $\mathbb{N}$ to $2^S$ but this is immediate since there is one from $\mathbb{N}$ to $S$.