Introduction to Mathematical Analysis I - Second Edition

50 2.5 LIMIT SUPERIOR AND LIMIT INFERIOR 2.4.3 Let f : [ 0 , ∞ ) → R be such that f ( x ) > 0 for all x . Define a n = f ( n ) f ( n )+ 1 . Prove that the sequence a n has a convergent subsequence. 2.4.4 Define a n = 1 + 2 n 2 n for n ∈ N . Prove that the sequence a n is contractive. 2.4.5 Let r ∈ R be such that | r | < 1. Define a n = r n for n ∈ N . Prove that the sequence { a n } is contractive. 2.4.6 Prove that the sequence 1 n ∞ n = 1 is not contractive. 2.5 LIMIT SUPERIOR AND LIMIT INFERIOR We begin this section with a proposition which follows from Theorem 2.3.1 . All sequences in this section are assumed to be of real numbers. Proposition 2.5.1 Let { a n } be a bounded sequence. Define s n = sup { a k : k ≥ n } (2.8) and t n = inf { a k : k ≥ n } . (2.9) Then { s n } and { t n } are convergent. Proof: If n ≤ m , then { a k : k ≥ m } ⊂ { a k : k ≥ n } . Therefore, it follows from Theorem 1.5.3 that s n ≥ s m and, so, the sequence { s n } is decreasing. Since { a n } is bounded, then so is { s n } . In particular, { s n } is bounded below. Similarly, { t n } is increasing and bounded above. Therefore, both sequences are convergent by Theorem 2.3.1 . Definition 2.5.1 Let { a n } be a sequence. Then the limit superior of { a n } , denoted by limsup n → ∞ a n , is defined by limsup n → ∞ a n = lim n → ∞ sup { a k : k ≥ n } . Note that limsup n → ∞ a n = lim n → ∞ s n , where s n is defined in ( 2.8 ) . Similarly, the limit inferior of { a n } , denoted by liminf n → ∞ a n , is defined by liminf n → ∞ a n = lim n → ∞ inf { a k : k ≥ n } . Note that liminf n → ∞ a n = lim n → ∞ t n , where t n is defined in ( 2.9 ) .