1. ## descending sequence proof

Hi, I need this problem to study off of. It is not for a grade. Please provide a complete prove if you can. Thanks,

Suppose $\displaystyle K_1 \supset K_2 \supset K_3 \supset$... is a descending sequence of nonempty sonnected subsets of a metric space M. Prove: if M is compact, the intersection $\displaystyle \bigcap K_n$ is a connected subset of M.

Also give a counterexample to show this is false without compactness.

Thanks.

2. Originally Posted by ElieWiesel
Hi, I need this problem to study off of. It is not for a grade. Please provide a complete prove if you can. Thanks,

Suppose $\displaystyle K_1 \supset K_2 \supset K_3 \supset$... is a descending sequence of nonempty sonnected subsets of a metric space M. Prove: if M is compact, the intersection $\displaystyle \bigcap K_n$ is a connected subset of M.
See this thread for some hints on how to do a similar problem.

Originally Posted by ElieWiesel
Also give a counterexample to show this is false without compactness.
In the space $\displaystyle \mathbb{R}^2$ let $\displaystyle A_n = \{(x,y):|x|<1,\ y<n\}$, and let $\displaystyle K_n$ be the complement of $\displaystyle A_n$, $\displaystyle K_n = \mathbb{R}^2\setminus A_n$. Then $\displaystyle K_n$ is the whole of $\displaystyle \mathbb{R}^2$ except for a strip of width 2 around the y-axis going from $\displaystyle -\infty$ up to the height y=n. This gives a descending sequence of nonempty connected subsets $\displaystyle K_n$ whose intersection is the whole of $\displaystyle \mathbb{R}^2$ except for the entire strip of width 2 around the y-axis. That is clearly not a connected set.

3. Edit: Never mind, I gave an incorrect proof.