#proofinatoot that monotone bounded sequences converge:

Let \((a_n)\) be an increasing bounded sequence; let \(A\) be the set of terms. Then \(A\) is bounded, so \(a=\sup A\) exists. For \(\epsilon > 0\), since \(a\) is the least upper bound, there exists \(N \in \mathbb{N}\) so that \(a_N > a-\epsilon\). Since \((a_n)\) is increasing, this means \(a_n > a-\epsilon\) for all \(n>N\); equivalently, \(|a_n - a|<\epsilon\), so \(\lim a_n = a\).

A similar argument applies to decreasing sequences.