"Let [tex]f:[a,b]\rightarrow [a,b][/tex] be defined such that [tex]|f(x)-f(y)|\leq a|x-y| [/tex] where 0<a<1. Prove that f is uniformly continuous and (other stuff)." Let e>0 and let d=e/a. Whenever [tex]0<|x-y|<d, |f(x)-f(y)|\leq a|x-y|<ad=e[/tex]. f is therefore by definition uniformly continuous. Did I do this right? It seems too good to be true. It doesn't seem right because I did not use the fact that 0<a<1.