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**flower3** show that sup S =1 where S = $\displaystyle \{ (-1)^{n} - \frac {1}{n^2} ; n \in N \} $

proof:

1) $\displaystyle \{ (-1)^{n} - \frac {1}{n^2} ; n \in N \} \to 1 \geq s , \forall s \in S$

2) if v< 1 want to prove $\displaystyle \exists s \in S $ such that $\displaystyle v<s $

$\displaystyle 1-v>0 \to \exists N^2 \ such \ that \ \frac {1}{N^2} < 1-v $ (by using Archimedean property ) "this step true or not ??"

$\displaystyle v < 1- \frac {1}{N^2} , \ \forall \ N \ is \ even $