There are ways of creating monotone sequences out of any sequence, and in this fashion we get the so-called limit superior and limit inferior. These limits always exist for bounded sequences. If a sequence \(\{ x_n \}\) is bounded, then the set \(\{ x_k : k \in \N \}\) is bounded.
10.2) Claim: Every bounded decreasing sequence is convergent. Proof: Let (s n) be a bounded decreasing sequence. Let S= fs n jn2Ng. By assumption, Sis a bounded set. By a corollary to the Completeness Axiom, Shas an in mum which is a real number. Let u= inf S. We now show that (s n) converges to u. Given >0, u+ is not a lower bound for S, since