An injection from $\mathbb{N}$ to $\mathbb{N}^n$. I'm currently attempting to prove $\mathbb{N}^n \sim \mathbb{N}$ via Cantor-Schroeder-Berstein (because I found no other way). In my work so far I've managed to find an injective function $f$ from $\mathbb{N}^n$ to $\mathbb{N}$ where $(a_1, \cdots, a_n) \mapsto 2^{a_1}3^{a_3}\cdots\rho_n^{a_n}$ where $\rho_n$ is the n-th prime number. I believe this function $f$ to be injective thanks to the fundamental theorem of arithmetic. But I can't seem to find an injective function from $\mathbb{N}$ to $\mathbb{N}^n$.
Resuming, I have three questions:


*

*Are my workings on $f$ correct?, that is, is $f$ really an injection?

*Do you have any hints on finding an injective function $g$ from $\mathbb{N}$ to $\mathbb{N}^n$?

*Is there another, easier way of proving $\mathbb{N}^n \sim \mathbb{N}$.


There are many other and better ways, look for the equivalence of $\mathbb{N}$ and $\mathbb{N}\times \mathbb{N}$ there is nice argument on the square.
Also one can just define such a map
$$(n,m) \mapsto \frac{(n+m)(n+m+1)}{2}+m$$
