Introduction to Mathematical Analysis I - 3rd Edition

54 2.5 Limit Superior and Limit Inferior 2.4.3 Let f : [0,∞) →Rbe such that f (x) >0 for all x. Define an = f (n) f (n)+1 . Prove that the sequence {an}has a convergent subsequence. 2.4.4 Define an = 1+2n 2n for n∈N. Prove that the sequence {an}is contractive. 2.4.5 Let r ∈Rbe such that |r| <1. Define 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 Rdefined in Definition 1.5.4. Along with the usual inequalities inR, we use c <∞, −∞<c for all c ∈R, and −∞<∞. Definition 2.5.1 Let {xn}be a sequence inRand let ℓ be a real number. We define the following: (i) limn→∞xn =ℓ if for any ε >0, there exists N∈Nsuch that xn ∈Rand |xn −ℓ| <ε for all n≥N. (ii) limn→∞xn =∞if for any M∈R, there exists N∈Nsuch that M<xn for all n≥N. (iii) limn→∞xn =−∞if for any M∈R, there exists N∈Nsuch that xn <Mfor all n≥N. Let {an}be a sequence of real numbers. Define sn =sup{ak : k ≥n} (2.8) and tn =inf{ak : k ≥n}. (2.9) Observe that in general {sn}and{tn}are sequences inR. Definition 2.5.2 Let {an}be a sequence of real numbers. Then the limit superior of {an}, denoted by limsupn →∞ an, is defined by limsup n→∞ an =lim n→∞ sn, where sn is defined in (2.8). Similarly, the limit inferior of {an}, denoted by liminfn→∞an, is defined by liminf n→∞ an =lim n→∞ tn, wheretn is defined in (2.9).

RkJQdWJsaXNoZXIy NTc4NTAz