Set theory cardinality question

Can anyone please give a really explicit proof (omitting no steps) and with as simple words as possible that any infinite set can be writtern as the union of disjoint countable sets?

Thank you.

Every infinite set X has a countable subset (this is a theorem of ZFC). We'll construct a function f from an initial set of ordinals to disjoint countably infinite subsets of X such that X its range. Let f(0) be any countably infinite subset of $X = X_0$. Then define $X_{i+1} = X_i \setminus f(i)$, $X_i = \displaystyle{ \cap_{j<i} X_j}$ for a limit ordinal i and let $f(i)$ is a countably infinite subset of $X_i$ (the possibility of making these choices collectively relies on the Axiom of Choice). The sequence must eventually end with a null set - otherwise, the set X would be larger than all cardinals.