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.

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# Need explanation of theorems on Uniform continuity

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