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Unformatted text preview: George and Martha Scrutinize the Deﬁnition of a Limit Math 8 2009, Chris Lyons M. Good news, my friend! “limn→∞ an = L.” G. What does that mean? M. It means, “The terms of the sequence {an } get closer and closer to L.” G. Well, what does “closer and closer” mean? M. It means, “The distance between L and the term an becomes smaller and smaller.” G. Okay, but how do you measure the distance between an and L? M. Using the absolute value of their diﬀerence. So I mean, “The quantity an − L becomes smaller and smaller.” G. How small does this quantity an − L actually become though? M. It becomes as small as you want. Smaller than any positive number you can think of! G. So you’re saying that if I give you any positive number, let’s call it... hmm... shall we call it ε? So you’re saying that if I give you any number ε > 0, then—no matter how small ε is!—the quantity an − L becomes smaller than ε? M. That’s right: “Given any ε > 0, the quantity an − L becomes smaller than ε.” G. What do you mean by “becomes” smaller? Becomes smaller as time goes on, or something...? M. No, I mean it becomes smaller as n gets bigger. So, “Given any ε > 0, as n grows larger, the quantity an − L becomes smaller than ε.” G. Wait, okay. Let’s say I’ve given you this ε. You say “as n grows larger”... but how large does n have to get before I know for sure that an − L is smaller than ε? M. By “as n grows larger,” I just mean, “as n passes a certain point,” or, “as soon as n is bigger than a certain natural number N .” G. Okay, let me see if I get this... you’re saying, “Given any ε > 0, there is some natural number N (which may depend on what ε is!) so that the following holds: whenever n ≥ N , we’re guaranteed to have an − L < ε.” Is that right? M. Precisely! 1 ...
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.
 Spring '08
 Ramakrishnan
 Math, Calculus, Linear Algebra, Algebra

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