limitgame

# limitgame - 6 = L this could mean that either a n has some...

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The Limit Game Math 8 2009, Chris Lyons Let { a n } be a sequence of real numbers, and let L R . This is a two-player game, and it’s played like this: 1. Player A gives Player B a (small) number ε > 0. 2. Player B tries come up with some N 1 so that the following is true: Whenever n N , we have | a n - L | < ε . (Put another way, if one wants to guarantee that | a n - L | < ε , one just needs to take n N .) 3. If Player B can give such an N , then they win. 4. If Player B can’t accomplish this, then Player A wins. Deﬁnition. If Player B can always win, then we say lim n →∞ a n = L . If there is some ε > 0 which will allow Player A to win, then lim n →∞ a n 6 = L . Note: if we have lim n →∞ a n
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Unformatted text preview: 6 = L , this could mean that either { a n } has some other limit, or that it has no limit at all. So in the langauge of this “game”, what does it mean to give a proof that lim n →∞ a n = L using the deﬁnition? It means giving a strategy which will allow Player A to always win the game; and, most importantly, it also means showing that the strategy will always work! Figuring out such a winning strategy can be hard. The good news, though, is that there’s often more than one winning strategy. So use your ingenuity, just like you would if you were trying to win a game! 1...
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