How many functions are there from $\mathbb Z$ to $\mathbb Z$? Repeating the question,

How many functions are there from $\mathbb Z$ to $\mathbb Z$?

A function $f \colon A \to B$ is a subset of $A \times B$ satisfying
$$(a,b) = (a,c) \qquad \Rightarrow \qquad b = c,$$
so it's enough (maybe) to look at subsets of $\mathbb Z \times \mathbb Z$. We know $|\mathbb Z \times \mathbb Z| = |\mathbb Z|$, and that the number of subsets of $\mathbb Z$ is $2^{|\mathbb Z|}$, but this counts finite subsets as well. Unsure of where to proceed from here.

Suppose $a,b,c,d,e,f,g,\ldots$ is an infinite sequence of positive integers (just use all of the infinitely many letters of the alphabet . . . . . ), thus a function from $\mathbb N$ to $\mathbb N$, and consider the number
$$
a + \cfrac 1 {b + \cfrac 1 {c + \cfrac 1 {d+ \cfrac 1 {e + \cfrac 1 {f + \cfrac 1 {g + \cdots}}}}}}.
$$
This gets you a one-to-one correspondence between the set of all functions from $\mathbb N$ to $\mathbb N$ and the set of all positive irrational numbers.
Now show that there is a one-to-one correspondence between the set of all functions from $\mathbb N$ to $\mathbb N$ and the set of all functions from $\mathbb Z$ to $\mathbb Z$ (that's easy) and also a one-to-one corresponence between the set of all positive irrational numbers and $\mathbb R$, and then you've got it.
