Partition of an infinite set into finitely many infinite sets?

Question

I know that an infinite set can be partitioned into 2 infinite subsets.

Can one partition an infinite set into finitely many infinite subsets?

Thanks

Answer 1

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

Answer 2

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. $\square$