My friend and I were discussing the cardinality of the countable cartesian product of countable sets. This is clearly bijective to the set of ALL rational sequences. We couldn't figure out the cardinality of this set. I say it's $\displaystyle \alpeh_1$, while he thinks that it may be $\displaystyle \aleph_2$.

It is clear that the set of rational Cauchy sequences is $\displaystyle \aleph_1$, but this is merely a subset of All rational sequences.

What is the cardinality of the set of All rational sequences? I would appreciate a sketch of a proof of this. Thanks in Advance.