(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

Since f is uniformly continuous, [tex]\left|f(x)-f(y)\right|< \epsilon[/tex] if [tex]\left|x-y\right|< \delta[/tex]. 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

[tex]\left|x-y\right|< \\left|M-P\right|<[/tex]. 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 [tex] \delta = abs(M-P)[/tex], 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.

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# Quick Proof rigor check.

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