An injection from $\mathbb{N}$ to $\mathbb{N}^n$.

Question

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:

1. Are my workings on $f$ correct?, that is, is $f$ really an injection?
2. Do you have any hints on finding an injective function $g$ from $\mathbb{N}$ to $\mathbb{N}^n$?
3. Is there another, easier way of proving $\mathbb{N}^n \sim \mathbb{N}$.

Answer 1

1. Yes.

2. The function $g(k)=(k,0,\ldots,0)$ should work. There are many injections from $\mathbb{N}$ in to the much larger, more spacious $\mathbb{N}^n$.

3. Once you know that $\mathbb{N}\sim\mathbb{N}^2$, you could work by induction.

Answer 2

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

$$(n,m) \mapsto \frac{(n+m)(n+m+1)}{2}+m$$