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Sequences - math.ucdavis.edu

Chapter 3. Sequences In this chapter, we discuss Sequences . We say what it means for a sequence to converge, and define the limit of a convergent sequence . We begin with some preliminary results about the absolute value, which can be used to define a distance function, or metric, on R. In turn, convergence is defined in terms of this metric. The absolute value Definition The absolute value of x R is defined by (. x if x 0, |x| =. x if x < 0. Some basic properties of the absolute value are the following. Proposition For all x, y R: (a) |x| 0 and |x| = 0 if and only if x = 0;. (b) | x| = |x|;. (c) |x + y| |x| + |y| (triangle inequality);. (d) |xy| = |x| |y|;. Proof. Parts (a), (b) follow immediately from the definition.)

Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and de ne the limit of a convergent sequence. We begin with some preliminary results about the absolute value, which can be used to de ne a distance function, or metric, on R. In turn, convergence is de ned in terms of this metric. 3.1. The ...

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