Are there uncountably many injective functions from $\mathbb N$ to $\mathbb N$? I think there are, but I haven't been able to prove this. I tried to make two injections, but I get stuck on trying to map all functions f from $\mathbb N$ to $\mathbb N$ onto the injective ones. How do you make sure f becomes injective, while making sure that the bigger injection stays injective? Or am I wrong and there are not even uncountably many functions?

Each injection from $\mathbb{N} \to \mathbb{N}$ can be thought of as an infinite sequence. Now use Cantor's diagonal argument to show uncountably many functions. 
