ask Baby Rudin chapter 3 exercise 4
Split $\{s_n\}$ into odd and even terms and we obtain two convergent sequences. If $\{t_n\}$ is a convergent subsequence of $\{s_n\}$, then since $\{t_n\}$ contains infinitely many even terms or odd terms, it converges to the limit of even sequence or odd sequence, respectively, by remark after definition 3.5. Hence the set $E$ defined as in definition 3.16 contains only two numbers, i.e., the limit of even sequence and the limit of odd sequence.