Let cn n 1 be a sequence of real numbers if for every

Question

Prove the lemma.

Let {c_n, n ≥ 1} be a sequence of real numbers. If for every subsequence there exists a further subsequence with a limit superior that does not exceed some number, a, say, then lim sup_n→∞ c_n ≤ a. Similarly, if for every subsequence there exists a further subsequence with a limit inferior that exceeds some number, b, say, then lim inf_n→∞ c_n ≥ b.