Let's say I'm trying to find a bijection from $\mathbb N^\mathbb N$, i.e., the set of all functions from $\mathbb N$ to $\mathbb N$, to some other set, say an open interval $(a,b)∈R$. What do I need to establish, to say there can exist a bijection? I do understand that I need to prove it's both injective and surjective, but how do you do it with a set of functions as the domain?
Let's say I'm trying to see if the injective part is true. So, I need to find a function that takes every possible sequence of natural numbers as input, and associates each of them with one real number in the interval $(a,b)$. But what does it mean to have a sequence of natural numbers as the input of a function? And even if that makes sense, what sort of functions can satisfy such a purpose? I hope I sound sane.