Partition of an infinite set into finitely many infinite sets?

Well, if you can do it with 2, you can keep partitioning one of them.

Of course. We do it by induction.

Theorem: If $A$ is an infinite set and $n>0$ is a natural number then we can write $A$ as a disjoint union of $n$ infinite sets.

Proof. For $n=1$ this is obvious, so we actually start with $n=2$.

For $n=2$ we can do it because every infinite set can be split into two infinite sets.

Suppose that we can split $A$ into $n$ parts, $A_1,\ldots, A_n$. Each is infinite, split $A_n$ into $B$ and $C$, and so the partition $A_1,\ldots,A_{n-1},B,C$ is a partition of $A$ into $n+1$ parts. $\square$