# A question to Sequence.

1. Mar 2, 2008

### Ka Yan

There is a theorem: If {En} is a sequence of closed, nonempty and bounded sets in a complete metric space X, if En$$\supset$$En+1, and if lim diam En = 0, then $$\cap$$En consists exactly one point.

And what I'm asking is that, if either the sets were not closed or X was not a complete space (but not both), and all other condictions are still satisfied, then what will follow? And if I let X be the rational set, for instance, what will I get. And could you explain it?

Thks.

Last edited: Mar 2, 2008
2. Mar 3, 2008

### CompuChip

Do you have the proof for this theorem? Then you could just scan through it and scrutinize each step to see which assumption(s) are used.

3. Mar 3, 2008

### tiny-tim

Hi, Ka Yan!

You should be able to find a simple example of open sets (on a plane, say) whose intersection is empty.

… there you go!

(and: hint: are the rationals complete? if not, why not?)