Let (Tn) be a bounded sequence. For all n ∈ N, define Sn = supTn, Tn 1, ..., Tk and tn = infTn, Tn 1, ..., Tk, where k = 2n.Prove that (Sn) is a monotone bounded sequence.It follows similarly that (tn) is a monotone bounded sequence. You do not have to prove this. Prove that tn < Sn for all n, m ∈ N. Prove that the sequences (Sn) and (tn) are convergent and that lim tn = lim Sn as n approaches infinity. Prove that lim tn = lim Sn if and only if the sequence (En) is convergent. (Hint: For 3,
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