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
∃ϵ>0∀δ>0∃x,y∈X |
In particular by letting we can construct two sequences and such that
Since is compact the two sequence have convergent subsequences i.e.
Since we have . Being continuous we hence conclude which is a contradiction
being .
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 |