can real line be written as a disjoint unions of set with cardinality 5 Can the real line be written as a disjoint union of sets with cardinality 5?
I tried using to write it as a set of sets.. but I didn't end up to a good position.

Let $\mathfrak c=|\mathbb R|$. Let $S_0=\{1,2,3,4,5\}$. We may recursively define $S_\alpha$ for $\alpha<\mathfrak c$ such that $S_\alpha$ has $5$ real numbers not in $\bigcup _{\beta<\alpha} {S_\beta}$, using the fact that $|\bigcup _{\beta<\alpha} {S_\beta}|=|\alpha\times 5|\leq \max(\omega,|\alpha|)<\mathfrak c.$
