Introduction to Mathematical Analysis I - Second Edition

54 2.5 LIMIT SUPERIOR AND LIMIT INFERIOR Theorem 2.5.10 Suppose { a n } is a sequence such that a n > 0 for every n ∈ N and liminf n → ∞ a n + 1 a n = ` > 1 . Then lim n → ∞ a n = ∞ . Example 2.5.1 Given a real number α , define a n = α n n ! , n ∈ N . When α = 0, it is obvious that lim n → ∞ a n = 0. Suppose α > 0. Then limsup n → ∞ a n + 1 a n = lim n → ∞ α n + 1 = 0 < 1 . Thus, lim n → ∞ a n = 0. In the general case, we can also show that lim n → ∞ a n = 0 by considering lim n → ∞ | a n | and using Exercise 2.1.3 . Exercises All sequences in this set of exercises are assumed to be in R . 2.5.1 Find limsup n → ∞ a n and liminf n → ∞ a n for each sequence. (a) a n = ( − 1 ) n . (b) a n = sin n π 2 . (c) a n = 1 +( − 1 ) n n . (d) a n = n sin n π 2 . 2.5.2 For a sequence { a n } , prove that: (a) liminf n → ∞ a n = ∞ if and only if lim n → ∞ a n = ∞ . (b) limsup n → ∞ a n = − ∞ if and only if lim n → ∞ a n = − ∞ . 2.5.3 Let { a n } and { b n } be bounded sequences. Prove that: (a) sup k ≥ n ( a n + b n ) ≤ sup k ≥ n a k + sup k ≥ n b k . (b) inf k ≥ n ( a n + b n ) ≥ inf k ≥ n a k + inf k ≥ n b k . 2.5.4 I Let { a n } and { b n } be bounded sequences. (a) Prove that limsup n → ∞ ( a n + b n ) ≤ limsup n → ∞ a n + limsup n → ∞ b n . (b) Prove that liminf n → ∞ ( a n + b n ) ≥ liminf n → ∞ a n + liminf n → ∞ b n . (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? 2.5.5 Let { a n } be a convergent sequence and let { b n } be an arbitrary sequence. Prove that (a) limsup n → ∞ ( a n + b n ) = limsup n → ∞ a n + limsup n → ∞ b n = lim n → ∞ a n + limsup n → ∞ b n . (b) liminf n → ∞ ( a n + b n ) = liminf n → ∞ a n + liminf n → ∞ b n = lim n → ∞ a n + liminf n → ∞ b n .