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?

Let $B = [0,1] \setminus \mathbb{Q}$ be the set of irrational positive numbers smaller than $1$.  
For each $b \in B$, let $(b_n)_{n\in \mathbb{N}}$ be its binary representation. i.e.
$$b = 0.b_0b_1b_2\cdots = \sum_{n=0}^\infty \frac{b_n}{2^{n+1}}\quad\text{ with all }\quad b_n \in \{ 0, 1 \}$$
Let $f_b : \mathbb{N} \to \mathbb{N}$ be the function defined by
$$f_b(n) = n + \sum_{k=0}^n b_k$$
It is easy to see $f_b(n)$ is strictly increasing in $n$, so each $f_b$ is injective.
Since different $b$ leads to different binary representation and hence different $f_b$.
The map 
$$B \ni b\quad\mapsto\quad f_b \in \{ f : \mathbb{N} \to \mathbb{N} \mid f \text{ injective } \}$$
is injective. Since $B$ has uncountably infinitely many elements, so does the set of injective functions over $\mathbb{N}$.
