Strategy_for_N

Strategy_for_N - Finding N : a vague outline Math 8 2009,...

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Finding N : a vague outline Math 8 2009, Chris Lyons Let’s suppose you have a sequence { a n } of real numbers, and you’re trying to prove that lim n a n = L by using the deﬁnition of the limit. Your proof reads as follows: Proof. Suppose that ε > 0 is given. Choose N to be ... and that’s as far as you’ve gotten. How do you ﬁnd the right N ? In general, this is a tough question to answer and it takes practice to ﬁnd the right N in diﬀerent situations. But here is a vague strategy on how to proceed. (You can also see the example below, which ought to help clarify the meaning of the steps.) 1. Take the quantity | a n - L | and try to simplify it as much as possible. If you’re able to remove the absolute value signs somehow, that also helps. 2. Try to ﬁnd some (positive) function g ( n ) which has both of the following properties: (a) It’s easy to show that there exists some N 1 such that n N 1 = g ( n ) < ε. (b) There exists some N 2 such that n N 2 = ⇒ | a n - L | ≤ g ( n ) . 3. Choose

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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.

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Strategy_for_N - Finding N : a vague outline Math 8 2009,...

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