How do I prove that $|\Bbb N^\Bbb N|$ is equal to $|2^\Bbb N|$?

Question

How do I prove that $|\Bbb N^\Bbb N|$ is equal to $|2^\Bbb N|$? I am very new to this subject an am looking for a solution with explanation. Thanks.

Answer 1

Hint:$$|2^\mathbb N|\leq|\mathbb N^\mathbb N|\leq\big|\big(2^\mathbb N\big)^\mathbb N\big|=|2^{\mathbb N\times\mathbb N}|=|2^\mathbb N|$$

The trickiest part of this statement to prove is $|\mathbb N^\mathbb N|\leq\big|\big(2^\mathbb N\big)^\mathbb N\big|$. We can construct an injection from the former to the latter as follows:$$\phi:\mathbb N^\mathbb N\to\big(2^\mathbb N\big)^\mathbb N$$$$f\mapsto g$$where $g(n)$ is the function $h_n:\mathbb N\to\{0,1\}$ defined as follows $$h_n(m)=\begin{cases}0&m=f(n)\\1&m\neq f(n)\end{cases}$$