Here is an easier proof that the cardinality is continuum. (Perhaps this is what Gowers was suggesting.)
First, any Cauchy sequence is equivalent to any of its subsequences. Second, if the original sequence has no repetitions, then there are continuum many such subsequences, since every subsequence is determined by an infinite subset of the indices, and there are continuum many such subsets. Finally, every Cauchy sequence is easily seen to be equivalent to a Cauchy sequence with no repetitions. Viola!