# Need explanation of theorems on Uniform continuity

by VreemdeGozer
Tags: continuity, explanation, theorems, uniform
 P: 12 I'm taking my first course in Analysis, and we learned a couple of theorems about Uniform Continuity. I have been able to visualize most of what's been going on before, but I need some help with the following: E $\subseteq$ ℝ, f: E $\rightarrow$ ℝ uniform continuous. if a sequence xn is Cauchy $\Rightarrow$ f(xn) is Cauchy I is a closed, bounded interval, f: I $\rightarrow$ ℝ. if f is continuous on I $\Rightarrow$ f is uniformly continuous on I We are using the international version of: An Introduction to Analysis by William R. Wade, fourth edition. I'm really looking for a visual explanation, but if anyone can explain why it works in words, that's fine too.