contractive maps are uniformly continuous
Proof Let be a contraction mapping in a metric space with metric . Thus, for some , we have for all ,
To prove that is uniformly continuous, let be given. There are two cases. If , our claim is trivial, since then for all ,
On the other hand, suppose . Then for all with , we have
In conclusion, is uniformly continuous.
The result is stated without proof in , pp. 221.
- 1 W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc., 1976.
|Title||contractive maps are uniformly continuous|
|Date of creation||2013-03-22 13:46:28|
|Last modified on||2013-03-22 13:46:28|
|Last modified by||mathcam (2727)|