This is an example from Rotman's Homological Algebra (p.122, old edition).
Let $G$ be the direct product of countable copies of $\mathbb{Z}$ (say, group of sequences of integers).
Fix a prime $p$.
Let $S$ be the subgroup consisting of all those sequences in $G$ such that for every $k\ge 1$, all except finitely many entries of sequence are divisible by $p^k$.
It is then shown in book that this subgroup is not free (by some arguments).
It was shown that this subgroup is uncountable; I didn't understand this fact about cardinality. Any hint for this?