What is the cardinality of $\Bbb{N^N}$?

Question

What is the cardinality of $\Bbb{N^N}$?

my answer: $|\mathbb{R}|$ $=$$|2^\mathbb{N}|$ $\leqslant$ $|\mathbb{N}^\mathbb{N}|$ $\leqslant$ $|\mathbb{R}^\mathbb{N}|$ $=$ $|(2^\mathbb{N})^\mathbb{N}|$ $=$ $|2^{\mathbb{N}\times\ \mathbb{N}}|$ $=$ $|2^\mathbb{N}|$ $=$ $|\mathbb{R}|$

In the end $|\mathbb{N}^\mathbb{N}|$ $=$ $|\mathbb{R}|$

Is that okay?

Answer 1

Yes. The answer is fine.

(Incidentally my students just have this question in their homework)

Answer 2

Your answer is correct, though I would of added that you know that for cardinals $k\leq l,l\leq k\,\implies k=l$