How to determine countability of $\Bbb N \times \Bbb N \times \Bbb N$? Is $\Bbb N \times\Bbb N \times\Bbb N$ countable or not, where $\Bbb N$ is the infinite set of all natural numbers ?


Please explain how it is done.
Define $f:\mathbb{N}\times\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ by $f(m,n,l)=2^m3^n5^l$ and observe that $f$ is injective by the Fundamental Theorem of Arithmetic.

Notice how this can be extended to show that any finite number of copies of $\mathbb{N}$ is countable.
Hint: Let $(x,y,z) \in \mathbf{N}\times\mathbf{N}\times\mathbf{N}$, let $w=x+y+z.$ What do you know about each $w \geq 3$ in relation to $x,y,z$?