Need explanation of theorems on Uniform continuity

[VreemdeGozer]

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 x_n is Cauchy \Rightarrow f(x_n) 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.

 

[lurflurf]

f is continuous so |f(x+h)-f(x)|<epsilon(x)
epsilon can vary with x
since I is closed and bounded max epsilon(x) exist thus
|f(x+h)-f(x)|<max epsilon(x)
hence f is uniform continuous

So the closed and bounded I allows us to deduce uniform continuity from continuity by limiting how much f can spread out.

 

[zooxanthellae]

I'll try for an intuitive idea of why these theorems are true. It's helpful that they're actually very, very similar ideas (if I am interpreting your statements correctly).

As is often true with theorems, it's helpful to understand why they're stated the why they are. Deleting hypotheses often helps us find counterexamples for the modified statement, and this will help us understand why the theorem is stated the way it is (and in turn make the statement more intuitive).

In the second example, we need a closed and bounded interval. This is good because that means any function continuous on that interval has a maximum and minimum. Hence we know that the distance between the function at any two points is bounded by, say, some number n. Given continuity, this is enough to get uniform continuity, since uniform continuity just requires that we can in some sense control the distance between the function at any two points by controlling the distance between the two points. It may be helpful to think about why we need a closed interval. To see why we want a closed interval, think about the topologist's sine curve. That would be problematic for uniform continuity. Conversely, what about if the interval was unbounded?

The Cauchy sequence bit is pretty similar. It helps to think about why we need it to be a Cauchy sequence. All that adds is that "after a while the rest of the terms are close together". But this is sort of like how we required that the interval in the above theorem be closed and bounded, so the logic is similar.

Of course these aren't proofs but I tried to give some of the intuition you seem to want. Hope it helps!