What is the cardinality of the set of bounded functions φ:N→N

Question

I know that the set of functions f:N→N is uncountable but I'm unable to find an injective function from an uncountable set to the set of bounded functions.

Answer 1

Hint:

• $2^\mathbb{N}$ is the set of functions $\mathbb{N} \to \{0, 1\}$.

I hope this helps $\ddot\smile$

Answer 2

As dtldarek has noted, there ae at least $2^{\aleph_0}$ such functions. But even if the functions weren't required to be bounded, there could be at most $\aleph_0^{\aleph_0}\le (2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}$ of them. Therefore, there are exactly $2^{\aleph_0}$ such functions. If you run through the proof of the Schröder–Bernstein theorem, you can even see how to construct an explicit bijection of $\mathcal{P}(\mathbb{N})$ with your set of functions.