Show that $| \mathbb N | = | \mathbb N \times \mathbb N |$, i.e., the cardinality of $\mathbb N$ is the same as the cardinality of $\mathbb N \times \mathbb N$.
How do I show it using the Fundamental Theorem of Arithmetic and the Schröder–Bernstein Theorem?
You should find injection from each set to other.
Let us first take a look at easy case, an injection $f\colon \mathbb{N}\rightarrow \mathbb{N}\times\mathbb{N}$. For instance take $$f(n)=(1,n).$$ Show for exercise it is really an injection.
Now we should find injection $g\colon \mathbb{N}\times\mathbb{N}\rightarrow \mathbb{N}$. Define $g$ as $$g(n,m)=2^n3^m.$$ Using fundamental theorem of arithmetic show that this is an injection.
By Cantor-Schröder-Bernstein theorem we get equality.
