54 2.5 Limit Superior and Limit Inferior 2.4.3 Let f : [0,∞) → R be such that f (x) > 0 for all x. Defne f (n) an = . f (n)+ 1 Prove that the sequence {an} has a convergent subsequence. 2.4.4 Defne 1 + 2n an = for n ∈ N. 2n Prove that the sequence {an} is contractive. 2.4.5 Let r ∈ R be such that |r| < 1. Defne an = r n for n ∈ N. Prove that the sequence {an} is contractive. ∞ 2.4.6 Prove that the sequence {1/n}n=1 is not contractive. 2.5 Limit Superior and Limit Inferior In this section, we consider the extended real line R defned in Defnition 1.5.4. Along with the usual inequalities in R, we use c < ∞, −∞ < c for all c ∈ R, and −∞ < ∞. Defnition 2.5.1 Let {xn} be a sequence in R and let ℓ be a real number. We defne the following: (i) limn →∞ xn = ℓ if for any ε > 0, there exists N ∈ N such that xn ∈ R and |xn − ℓ| < ε for all n ≥ N. (ii) limn →∞ xn = ∞ if for any M ∈ R, there exists N ∈ N such that M < xn for all n ≥ N. (iii) limn →∞ xn = −∞ if for any M ∈ R, there exists N ∈ N such that xn < M for all n ≥ N. Let {an} be a sequence of real numbers. Defne sn = sup{ak : k ≥ n} (2.8) and tn = inf{ak : k ≥ n}. (2.9) Observe that in general {sn} and {tn} are sequences in R. Defnition 2.5.2 Let {an} be a sequence of real numbers. Then the limit superior of {an}, denoted by limsupn →∞ an, is defned by limsupan = lim sn, n→∞ n →∞ where sn is defned in (2.8). Similarly, the limit inferior of {an}, denoted by liminfn →∞ an, is defned by liminfan = lim tn, n→∞ n→∞ where tn is defned in (2.9).
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