Define cauchy sequences

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Math 104: Homework 3-

1. Let (an) be a sequence defined according to a1 = t where t > 0, and an+1 = 2an/(1+an) for n ∈ N. Prove that an → 1 as n → ∞.

2. Let (sn) and (tn) be Cauchy sequences defined on R, and let (un) be a sequence defined as un = asn + btn for all n, where a, b ∈ R. By using the definition of a Cauchy sequence only, without assuming that limits of (sn) and (tn) exist, prove that (un) is a Cauchy sequence.

3. Let (sn) be a sequence defined by sn = (-1)n(1 + 1/n). Prove that lim sup sn = 1 and lim infsn = -1.

Reference no: EM131081498

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