10.2 Theorem. All bounded monotone sequences converge. Proof Let (sn) be a bounded increasing sequence. Let S denote the set {sn n € N}, and let u = sup S. Since S is bounded, u represents a real number. We show lim sn = u. Let € > 0. Since u — e is not an upper bound for S, there exists N such that sy > u − e. Since (sn) is increasing, we have sã ≤ sʼn for all n ≥ N. Of course, sn ≤ u for all n, so n > N implies u − € < Sn ≤ u, which implies |sn − u| < €. This shows lim sñ = u. The proof for bounded Exercise 10.2. decreasing sequences is left to

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