Proof of Cantor's Theorem

Theorem

Letbe a countable family of closed nonempty subsets of a complete spacesuch that

and(the diameter of the setstends to 0)

Then

Proof

For everychoose

Giventhere existssuch that for

Forhence

Henceis a Cauchy sequence inSinceis complete,

Nowandis closed so

Thusand