sequence question !

**flower3** · October 30th 2009, 06:02 PM

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

**proscientia** · October 30th 2009, 07:20 PM

Originally Posted by flower3

Let S be a nonempty bounded above subset of R . prove that there is a sequence $x_n \color{red}\in\color{black} S$ of elements of S that converges to sup S

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

**flower3** · October 30th 2009, 11:40 PM

Originally Posted by proscientia

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

it's not belongs to !!!!
both $x_n \ and \ S$ are sets ,so the truth subsets

**HallsofIvy** · October 31st 2009, 01:30 AM

[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.

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