Let S be a nonempty bounded above subset of R . prove that there is a sequence $\displaystyle x_n \subseteq S$ of elements of S that converges to sup S

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- Oct 30th 2009, 06:02 PMflower3sequence question !
Let S be a nonempty bounded above subset of R . prove that there is a sequence $\displaystyle x_n \subseteq S$ of elements of S that converges to sup S

- Oct 30th 2009, 07:20 PMproscientia
Let $\displaystyle s=\sup S.$ Then $\displaystyle \forall\,n\in\mathbb N,$ $\displaystyle \exists\,x_n\in S$ such that $\displaystyle s\geqslant x_n>s-\frac1n$ since $\displaystyle s-\frac1n$ is not an upper bound for $\displaystyle S.$ The sequence $\displaystyle \left(x_n\right)_{n\,=\,1}^\infty$ converges to $\displaystyle s.$

- Oct 30th 2009, 11:40 PMflower3
- Oct 31st 2009, 01:30 AMHallsofIvy
[PHP][/PHP]

**You**said "a sequence of elements of S". Did you mean "a sequence of subsets of S"? sup S is definitely a number. How does a sequence of sets converge to a number? I think**you**have misunderstood the question.