Let an be a sequence such that the subsequences a2k a2k1 and

Let {a_n} be a sequence such that the subsequences {a_2k}, {a_2k+1}, and {a_3k} converge. Show that the sequence {a_n} also converges.

Solution

Given that {a_2k}, {a_2k+1} and {a_3k} are convergent subsequences.

We need to show that sequence {a_n} also coverge.

Let us find a new subsequence of {a_n} using the subsequences {a_2k} and {a_2k+1}. On adding these two subsequences, we get: {a_2k}+{a_2k+1}

This sequence basically covers all the values possible for sequence {a_k} because  {a_2k} covers all the even terms and {a_2k+1} covers all the odd terms. Therefore, this combined sequence is nothing but sequence {an} itself. And we know that sum of two convergent sequences is always convergent. Therefore, {an} is convergent!