Convergence Limit Theorems Monotone Sequences The Bolzano-Weierstrass Theorem Limit Superior and Limit Inferior 2. SEQUENCES We introduce the notion of limit frst 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, an = f (n). We will also denote sequences by {an} ∞ n=1, {an}n, or even {an}. Each value an is called a term of the sequence, more precisely, the nth term of the sequence. ■ Example 2.0.1 Consider the sequence {an} given by an = 1 for n ∈ N. This is a sequence of n rational numbers. On occasion, when the pattern is clear, we may list the terms explicitly as in 1 1 1 1 1, , , , ,... 2 3 4 5 ■ Example 2.0.2 Let {an} be the sequence given by an =(−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 Defnition 2.1.1 Let {an} be a sequence of real numbers. We say that the sequence {an} converges to a ∈ R if for any ε > 0 there exists a positive integer N such that |an − a| < ε for any n ∈ N with n ≥ N. In this case, we call a the limit of the sequence (see Theorem 2.1.1 below) and write limn →∞ an = a. If the sequence {an} does not converge, we call the sequence divergent.
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