I have to prove:
Let $\mathcal{F}$ be a non-trivial ultrafilter on $\mathcal{P}(\mathbb{N})$. Prove that the ultraproduct $ \mathbb{N}^* = {\mathbb{N}^{\mathbb{N}}}/{\mathcal{F} } $ (I don't know if this is standard notation) is uncountable.
HINT: Prove there exist a function $F:\mathbb{N}^\mathbb{N} \rightarrow \mathbb{N}^\mathbb{N} $ such that for all $f,g \in \mathbb{N}^\mathbb{N}$, if $f \neq g$: $\exists n \forall m > n[(F(f))(m) \neq (F(g))(m)] $
In the hint, the first $f$ should be $F$ to match the notation later. I suggest you define $F$ so that, for each function $f\in\mathbb N^{\mathbb N}$ and each natural number $n$, $F(f)(n)$ encodes the finite sequence $\langle f(0),f(),\dots,f(n)\rangle$.