Formal Proof of Uniform Continuity on a Closed Interval

[MathSquareRoo]

Homework Statement

Prove that if f is uniformly continuous on [a,b] and on [a,c] implies that f is uniformly continuous on [a,c].

Homework Equations

The Attempt at a Solution

This is my rough idea for a proof, can someone help be say this more formally? Is my thinking even correct?

Let epsilon > 0.
Then there is some delta_1 for [a,b] and some delta_2 for [b,c].
Then the minimum of delta_1 and delta_2 is the delta we want for [a,c].

 

[Dick]

Homework Statement

Prove that if f is uniformly continuous on [a,b] and on [a,c] implies that f is uniformly continuous on [a,c].

Homework Equations

The Attempt at a Solution

This is my rough idea for a proof, can someone help be say this more formally? Is my thinking even correct?

Let epsilon > 0.
Then there is some delta_1 for [a,b] and some delta_2 for [b,c].
Then the minimum of delta_1 and delta_2 is the delta we want for [a,c].

Sure, that's the idea. It's not that hard to fill that out to a formal proof.

 

[MathSquareRoo]

Thanks. What changes should I make to make it more formal?

 

[Dick]

Thanks. What changes should I make to make it more formal?

Just fill in some words. "there is some delta_1 for [a,b]" doesn't mean much. There is some delta_1 for [a,b] such that what? I know what you mean, but spell it out.