Let (Kn n N) be a sequence of nonempty compact sets in R such that K1 K2 Kn Prove that there exists at least one point x R such that x Kn for all n N, that is, the intersection is not empty Kn...

For each n N let x n k n Since the set x n K 1 we infer ...

Let (Kn : n N) be a sequence of nonempty compact sets in R such that K1

Question:

Let (Kn : n ˆˆ N) be a sequence of nonempty compact sets in R such that K1 Šƒ K2 Šƒ ˆ™ ˆ™ ˆ™ Kn Šƒ ˆ™ ˆ™ ˆ™. Prove that there exists at least one point x ˆˆ R such that x ˆˆ Kn for all n ˆˆ N, that is, the intersection is not empty.