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.

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