Partitioning $\Bbb{N}$

Question

Can we partition $\Bbb{N}$ into a finite union $$J_1 \sqcup J_2\ldots \sqcup J_N= \Bbb{N}$$

where $\sqcup$ denotes disjoint union. I'm guessing if we can then one of the $J_i$ must be infinite and only one otherwise we get a contradiction right?

Answer 1

Take $J_1$ the odd numbers and $J_2$ the even ones - Or, take $J_1$ the odds, $J_2$ those divisible by 2 and not by $4$ and $J_3$ those divisible by 4 - All of these sets are infinite.

Answer 2

Certainly a finite union of finite sets is finite, by if you pick your $J_1$ and $J_2$ to be even and odd numbers respectively, that is an example with two infinite sets.