1. MONOTONE SEQUENCES (a) Let X be a non-empty set of real numbers which is bounded above. Prove that there is a monotone increasing sequence of elements of X converging to sup(X). (b) Consider the sequence (2n) defined by and so on with n+1 (d) Prove that x1 = 1 23 = √₁+ and so on with n+1 = √I+. Prove that (,) converges and find the limit. (c) Consider the sequence (an) defined by 2₁=1 x2 = √1+1 /1+√1+1 2₂=1+ x3 = 1+ + 1+ 1+ 1+1+Prove that (n) converges and find the limit. 1 2 +...+ + 2√2 ≤3 1V1 2√2 3√3 n√n √n for all n 1. Prove that the sequence on the left side of the equation converges. (e) Let K> 1 be a real number. Consider the sequence #₁ = 1 and n+1 2K for all n ≥ 1. 2+K Prove that for all n we have a ≤K and an ≤n+1. Prove that (₂) converges to VK.

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