Theorem: If f(x) is continuous on [a,b], it is uniformly continuous.
Proof: Given ε, assume there is an x from [a,b] such that no matter how small I make δ,
│f(x) - f(p)│not < ε if │x - p │ < δ.
Then f(x) is not continuous at x → there is a δ independent of x and f(x) is uniformly continuous. Same wording applies for a compact set.
Why bother with a Covering Theorem?