3.10N_Epsilon-Delta_Proofs

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern