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:(adsbygoogle = window.adsbygoogle || []).push({});

E [itex]\subseteq[/itex] ℝ, f: E [itex]\rightarrow[/itex] ℝ uniform continuous. if a sequence x_{n}is Cauchy [itex]\Rightarrow[/itex] f(x_{n}) is Cauchy

I is a closed, bounded interval, f: I [itex]\rightarrow[/itex] ℝ. if f is continuous on I [itex]\Rightarrow[/itex] 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Need explanation of theorems on Uniform continuity

**Physics Forums | Science Articles, Homework Help, Discussion**