How to partition natural number $N$ into countable many of subsets each having countable number of element?

Question

The cardinality of "countable many of subsets each having countable number of element" is $|N\times N|$

It is known that $|N\times N|=|N|$, so such partition must be possible.

However, I am having a hard time finding this partition.

Answer 1

I think a simple enough approach is using a numbers' digits.

For example, you can define $A_k$ to be the set of all natural numbesr where the digit $1$ (written in base $10$) appear exactly $k$ times. You have $ \mathbb N = \cup _k A_k$ and $|A_k| = \aleph_0$ for each $k$.

Exercise: try to find a similar division using prime factors, which is also a useful approach in many cases.

Answer 2

My favorite approach to this question is the function $f:\mathbb{N}^2 \to \mathbb{N}$, where $f:(x,y) \mapsto 2^x(2y-1)$. Verifying that this is a bijection requires the fundamental theorem of arithmetic.