See this thread for some hints on how to do a similar problem.

In the space let , and let be the complement of , . Then is the whole of except for a strip of width 2 around the y-axis going from up to the height y=n. This gives a descending sequence of nonempty connected subsets whose intersection is the whole of except for the entire strip of width 2 around the y-axis. That is clearly not a connected set.