Convergence Limit Theorems Monotone Sequences The Bolzano-Weierstrass Theorem Limit Superior and Limit Inferior 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 Ais a function f : N→A. We will denote the image of nunder 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 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 {an}be the sequence given byan = (−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 {an}be a sequence of real numbers. We say that the sequence {an}converges toa∈Rif for any ε >0 there exists a positive integer Nsuch that |an −a| <ε for any n∈Nwithn≥N. In this case, we call athelimit 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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