Analysis of Limits Proof
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Analysis of Limits Proof

[From: ] [author: ] [Date: 11-10-02] [Hit: ]
Thanks =D-Assume that the limit of {a[n]} equals L 0. Then, by the definition of L, there exists M such that for any n>M,So,.......
Prove or disprove that if (x_n) is convergent and lim(x_n)<2 then ∃ M∈N and c>0 such that when n≥M, x_n<2-c

Any help would be greatly appreciated! Thanks =D

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Assume that the limit of {a[n]} equals L < 2. Let c = (2-L)/2 >0. Then, by the definition of L, there exists M such that for any n>M, |a[n] - L| < c, or equivalently, -c < a[n] - L < c.
So, consider a[n] - L < e:
a[n] < L + c = 1 + L/2 = 2 - (2-L)/2 = 2 - c.
.
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keywords: Analysis,Proof,Limits,of,Analysis of Limits Proof
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