proof of Heine-Cantor theorem

We prove this theorem in the case when X and Y are metric spaces.

Suppose f is not uniformly continuous. Then


In particular by letting δ=1/k we can construct two sequences xk and yk such that


Since X is compactPlanetmathPlanetmath the two sequence have convergent subsequences i.e.


Since d(xk,yk)0 we have x¯=y¯. Being f continuousPlanetmathPlanetmath we hence conclude d(f(xkj),f(ykj))0 which is a contradictionMathworldPlanetmathPlanetmath being d(f(xk),f(yk))ϵ.

Title proof of Heine-Cantor theorem
Canonical name ProofOfHeineCantorTheorem
Date of creation 2013-03-22 13:31:26
Last modified on 2013-03-22 13:31:26
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 5
Author paolini (1187)
Entry type Proof
Classification msc 46A99