You could try an argument by contradiction. Suppose the result is false, so that every such subsequence converges. In particular, the subsequences and (of even-numbered and odd-numbered terms terms in the sequence) both converge, say and . The subsequence also converges, and it's easy to see that its limit has to be both l_1 and l_2. So that means that , and the whole sequence has to converge to that limit.