Introduction to Mathematical Analysis I - Second Edition

CONVERGENCE LIMIT THEOREMS MONOTONE SEQUENCES THE BOLZANO-WEIERSTRASS THEOREM LIMIT SUPERIOR AND LIMIT INFERIOR OPEN SETS, CLOSED SETS, COMPACT SETS, AND LIMIT POINTS 2. SEQUENCES We introduce the notion of limit first through sequences. As mentioned in Chapter 1, a sequence is just a function with domain N . More precisely, a sequence of elements of a set A is a function f : N → A . We will denote the image of n under the function with subscripted variables, for example, a n = f ( n ) . We will also denote sequences by { a n } ∞ n = 1 , { a n } n , or even { a n } . Each value a n is called a term of the sequence, more precisely, the n -th term of the sequence. Example 2.0.1 Consider the sequence a n = 1 n for n ∈ N . This is a sequence of rational numbers. On occasion, when the pattern is clear, we may list the terms explicitly as in 1 , 1 2 , 1 3 , 1 4 , 1 5 , . . . Example 2.0.2 Let a n = ( − 1 ) n for n ∈ N . This is a sequence of integers, namely, − 1 , 1 , − 1 , 1 , − 1 , 1 , . . . Note that the sequence takes on only two values. This should not be confused with the two-element set { 1 , − 1 } . 2.1 CONVERGENCE Definition 2.1.1 Let { a n } be a sequence of real numbers. We say that the sequence { a n } converges to a ∈ R if, for any ε > 0, there exists a positive integer N such that for any n ∈ N with n ≥ N , one has | a n − a | < ε ( or equivalently, a − ε < a n < a + ε ) . In this case, we call a the limit of the sequence (see Theorem 2.1.3 below) and write lim n → ∞ a n = a . If the sequence { a n } does not converge, we call the sequence divergent . Remark 2.1.1 It follows directly from the definition, using the Archimedean property, that a sequence { a n } converges to a if and only if for any ε > 0, there exists a real number N such that for any n ∈ N with n > N , one has | a n − a | < ε .