Proof: Uniformly Continuous Function on a Bounded Set is Bounded

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Homework Statement

Let f be a real uniformly continuous function on the bounbed set E in R. Prove that f is bounded on E.

Homework Equations

The Attempt at a Solution

Since f is uniformly continuous, \left|f(x)-f(y)\right|< \epsilon if \left|x-y\right|< \delta. Since E is bounded, there exists some maximal distance between x and y; let M be the upper limit and P be the lower limit of E. Then

\left|x-y\right|< \\left|M-P\right|<. If M, P are not in E, then define two sequences {x_n} and {y_n} such that lim n->infinity {x_n}=M and lim n->infinity {y_n} = P.

Then the definition of uniform continuity implies that if we let \delta = abs(M-P), so that abs(x-y) < delta for all x, y then abs(f(x)-f(y))< epsilon for all f(x), f(y), with epsilon equal to lim n->infinity {f(x_n)}=M and lim n->infinity {f(y_n)}

I feel fairly solid about this, but I'm finding the difference between absolute continuity and continuity a bit confusing and would like a rigor/correctness check.

Thanks.

 

[phreak]

Here's where your proof goes wrong:

Then the definition of uniform continuity implies that if we let LaTeX Code: \\delta = abs(M-P) , so that abs(x-y) < delta for all x, y then abs(f(x)-f(y))< epsilon for all f(x), f(y), with epsilon equal to lim n->infinity {f(x_n)}=M and lim n->infinity {f(y_n)}

You can't just pick a delta like this. You can only pick an epsilon and then prove existence of delta using the definition of uniform continuity. Here's how I would prove it:

Choose arbitrary epsilon > 0. Then there exists delta>0 such that as long as |x-y|<delta, |f(x)-f(y)|<epsilon. Choose N so large that N delta > |M-P|. Now, choose arbitrary x0 in E. Then we must have f(x0) - epsilon N < f(x) < f(x0) + epsilon N for all x in E (check this). This proves the theorem.

 

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Hm, ok. So that is the same idea I think; or at least that makes complete sense and I was just being silly.

Thanks.