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Thread: convergence

  1. #1
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    convergence

    Construct a sequence {$\displaystyle x_{n}$} of reals such that a = $\displaystyle \sup{x_{n}:n\in N}$ exists, a is not the limit of $\displaystyle x_{n}$ and for every $\displaystyle \epsilon\ge 0$ and k a positive integer, there exists a positive integer N greater than k such that the absolute value of ($\displaystyle x_{n}-a$) is less than $\displaystyle \epsilon$
    Last edited by Plato; Apr 20th 2010 at 07:11 AM. Reason: messed up the latex
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  2. #2
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    Quote Originally Posted by janae77 View Post
    Construct a sequence {$\displaystyle x_{n}$} of reals such that a = $\displaystyle \sup{x_{n}:n\in N}$ exists, a is not the limit of $\displaystyle x_{n}$ and for every $\displaystyle \epsilon\ge 0$ and k a positive integer, there exists a positive integer N greater than k such that the absolute value of ($\displaystyle x_{N}-a$) is less than $\displaystyle \epsilon$
    Simplest example: $\displaystyle x_n=(-1)^n$
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