- #1

HumbleHarry

- 7

- 0

## Homework Statement

Let ##{X_n}## be a sequence of nonempty compact subsets of the metric space Ω such that ##{X_n+1}## ##\subseteq## ##{X_n}##, with ##n: 1→∞##. Prove that their intersection is nonempty.

**By the way, I mean to subscript "n+1", not have ##{X_n}## + 1 as it seems.

## Homework Equations

One can use sequential compactness to describe the compactness of each subset. Every sequence in ##{X_n}## has a subsequence ##{X_k}## where ##x_k##→##x_o## ##\in## ##{X_n}##.

## The Attempt at a Solution

I tried using sequential compactness to find some limit point that's found in ##{X_n}## and all of its subsets, but I have no clue on how to do so. I've tried setting up singleton sequences around the points in each nonempty set to say they must have a subsequence that converges to said point, implying said point is in ##{X_n}##, but I don't know if this implies that these points lie in the other sets, as well.

On another line of thought, since ##{X_n+1}## is a subset of ##{X_n}##, there's some sequence in ##{X_n}## that's also in ##{X_n+1}##, both of which should converge to a point in ##{X_1}##.

Help would be appreciated.