Cardinality of $\Bbb N^{k}$

Question

How can i determine the cardinality of $\Bbb N^{k}$ for $k \in \Bbb N$ ?

I know that $\Bbb{N} \times \Bbb{N}$ is of cardinality $\aleph_o$, is there any valid induction for $k\in \Bbb{N}$?

Answer 1

Yes. Let $f:\Bbb N^2\to \Bbb N$ be a bijection. For $k\geq 2$ suppose there is a bijection $g: \Bbb N^k\to \Bbb N.$ For $x=(x_1,...,x_{n+1})\in \Bbb N^{n+1}$ let $h(x)=f(g(x_1,...,x_n),x_{n+1}).$ Then $h:\Bbb N^{n+1}\to \Bbb N$ is a bijection.

This is a common technique in inductive proofs.