59 Exercises 2.5.1 Find limsupn →∞ an and liminfn→∞an for each sequence. (a) an =sin nπ 2 . (b) an = 1+(−1)n n . (c) an =nsin nπ 2 . 2.5.2 Let {an}and{bn}be bounded sequences. Prove that: (a) sup{ak +bk : k ≥n} ≤sup{ak : k ≥n}+sup{bk : k ≥n}. (b) inf{ak +bk : k ≥n} ≥inf{ak : k ≥n}+inf{bk : k ≥n}. 2.5.3 ▶Let {an}and{bn}be bounded sequences. (a) Prove that limsupn →∞(an +bn) ≤limsupn →∞ an +limsupn →∞ bn. (b) Prove that liminfn→∞(an +bn) ≥liminfn→∞an +liminfn→∞bn. (c) Find two counterexamples to show that the equalities may not hold in part (a) and part (b). Is the conclusion still true in each of parts (a) and (b) if the sequences involved are not necessarily bounded? 2.5.4 Let {an}be a convergent sequence and let {bn}be an arbitrary sequence. Prove that (a) limsupn →∞(an +bn) =limsupn →∞ an +limsupn →∞ bn =limn→∞an +limsupn →∞ bn. (b) liminfn→∞(an +bn) =liminfn→∞an +liminfn→∞bn =limn→∞an +liminfn→∞bn.
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