3.10N_Epsilon-Delta_Proofs

# 3.10N_Epsilon-Delta_Proofs - Math 280 Class Notes 3.10...

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Math 280 Strategies of Proof Fall 2007 Class Notes 3.10 Epsilon-Delta Proofs The definitions of the limit of an infinite sequence and of the limit of a function are among the most important in mathematics. They are also one of the greatest sources of di culty for mathematics students. In this section, we want to examine how to apply our proof-writing techniques to an “epsilon-delta proof.” We’ll start by examining proofs involving the limit of an infinite sequence. We’ll see that the same general procedures apply to proofs involving the definition of a limit of a function. Limits of Sequences We begin with the definition of a sequence. Definition. A sequence is a function s : N R whose domain is the set N of natural numbers. We usually denote the value of a sequence at n by s n instead of s ( n ). We also refer to a sequence s as ( s n ) or by listing the elements ( s 1 , s 2 , s 3 , . . . ). We can graph a sequence as a set of points in the cartesian plane (thinking of it as a function) or as a set of points on the number line. Example. Let ( s n ) be the sequence defined by s n = 1 1 n for all n N . The sequence can then be written as ( s n ) = ° 0 , 1 2 , 2 3 , 3 4 , . . . ¢ . In order to visualize the sequence, we can graph it in the Cartesian plane or on a number line. To graph the sequence in the Cartesian plane, we can view the sequence as a function from N into R . For each n N , there corresponds exactly one sequence value s n , which can be represented by the point ( n, s n ) in the Cartesian plane. The graph of this sequence then consists of the set of points © (1 , 0) , (2 , 1 2 ) , (3 , 2 3 ) , (4 , 3 4 ) , . . . . 1 2 3 4 5 6 1 n Since the domain of the function, N , is a discrete set of points, rather than a continuous interval, the graph of the sequence is itself a discrete set of points. To graph the sequence on a number line, we can plot a point for each sequence term s n . s 1 s 2 s 3 s 4 0 1 139

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140 Chapter 3 Proof Techniques For the sequence s n = 1 (1 /n ) in the preceding example, as n gets larger and larger, then 1 /n gets closer and closer to 0, which means the sequence terms get closer and closer to a value of 1. In the graph in the Cartesian plane, the points are approaching the horizontal line y = 1, while in the graph on the number line, the point are clustering near the value of 1. We describe this by saying that the sequence converges to 1, or that the limit of the sequence is 1. For this relatively simple example, we can see intuitively how the sequence behaves. For more complex sequences, an intuitive approach may not su ce. Furthermore, we need to obtain a more precise and rigorous mathematical description for terms such as “closer and closer” or “approaches.” This is furnished by the following definition. At first (or even second or third) reading, it may not be clear how this relates to our intuitive notion of convergence of a sequence, but after working with it for awhile, you should be able to see the connection.
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