Example of an Empty Intersection with Converging Nested Sets in Incomplete Space

[jessicaw]

Homework Statement

If {S_n} is a nest of decreasing
non-empty closed subsets of X with \lim_{n\to\infty}d(S_n)=0,
give an example of \cap_{n=1}^{\infty}S_n which is empty!

Homework Equations

no

The Attempt at a Solution

i know X cannot be R, further reading i know that X cannot be complete. But i cannot think of an incomplete space and a suitable S_n.

 

[micromass]

Any incomplete space should do. I suggest taking X=\mathbb{Q} and setting S_n=[\pi-1/n,\pi+1/n].