55 ■ Example 2.5.1 Consider the sequence {an} given by an =(−1) n . For any n ∈ N, sn = sup{ak : k ≥ n} = 1 and tn = inf{ak : k ≥ n} = −1. Then limn →∞ sn = 1 and limn →∞ tn = −1. Thus, limsupn →∞ an = 1 and liminfn →∞ an = −1. ■ Example 2.5.2 Consider the sequence {an} given by an =(−1) nn. For any n ∈ N, sn = sup{ak : k ≥ n} = ∞ and tn = inf{ak : k ≥ n} = −∞. Then limn →∞ sn = ∞ and limn →∞ tn = −∞. Thus, limsupn →∞ an = ∞ and liminfn →∞ an = −∞. ■ Example 2.5.3 Consider the sequence {an} given by an = n. For any n ∈ N, sn = sup{ak : k ≥ n} = ∞ and tn = inf{ak : k ≥ n} = n. Then limn →∞ sn = ∞ and limn →∞ tn = limn →∞ n = ∞. Thus,limsupn →∞ an = ∞ and liminfn →∞ an = ∞. In a similar way, if {bn} is given by bn = −n, we have limsupn →∞ bn = −∞ and liminfn →∞ bn = −∞. Proposition 2.5.1 Let {an} be a bounded sequence of real numbers. Then limsupn →∞ an and liminfn →∞ an exist (as real numbers). Proof: Since {an} is bounded, we see that the sequence {sn} defned in (2.8) is a bounded sequence of real numbers, so it is bounded below. If m ≤ n, then {ak : k ≥ n}⊂{ak : k ≥ m}. Thus, it follows from Theorem 1.5.3 that sn ≤ sm, so the sequence {sn} is decreasing. Similarly, the sequence {tn} defned in (2.9) is increasing and bounded above. Therefore, both sequences are convergent by Theorem 2.3.1. By the defnition, limsupn →∞ an and liminfn →∞ an exist (as real numbers). □ Proposition 2.5.2 Let {an} be a sequence of real numbers. If {an} is not bounded above, then limsupan = ∞. n→∞ Similarly, if {an} is not bounded below, then liminfan = −∞, n→∞ Proof: Suppose {an} is not bounded above. We will show that limn →∞ sn = ∞, where sn is defned in (2.8). Since {an} is not bounded above, for any n ∈ N, the set {ak : k ≥ n} is also not bounded above. Thus, sn = sup{ak : k ≥ n} = ∞ for all n. Therefore, limsupan = lim sn = ∞. n→∞ n →∞ The proof for the second case is similar. □ Proposition 2.5.3 Let {an} be a sequence of real numbers. Then (i) limsupn →∞ an = −∞ if and only if limn →∞ an = −∞, (ii) liminfn →∞ an = ∞ if and only if limn →∞ an = ∞.
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