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I'm kinda stuck on this problem. (Worried) Please help me. Thank you much.

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- Apr 15th 2009, 07:44 PMjohn_n82limit of an increasing sequence
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I'm kinda stuck on this problem. (Worried) Please help me. Thank you much. - Apr 16th 2009, 03:00 AMxalk

Since x is the supremum of {$\displaystyle x_{n}:n\in n$},

GIVEN ε>0 there exists kεN AND SUCH that:

$\displaystyle x-\epsilon< x_{k}\leq x<x+\epsilon$

But since {$\displaystyle x_{n}$} is increasing and bounded by x:

for all n $\displaystyle x_{n}\leq x$ and

for $\displaystyle n\geq k$ $\displaystyle x_{n}\geq x_{k}$

hence :

$\displaystyle x-\epsilon<x_{n}\leq x<x+\epsilon\Longleftrightarrow|x_{n}-x|<\epsilon$

THUS $\displaystyle \lim_{n\rightarrow\infty}{x_{n}} = x$ - Apr 16th 2009, 08:56 AMjohn_n82
- Apr 16th 2009, 09:03 AMPlato