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

Suppose that the function [itex]f|(a,b)→ℝ[/itex] is uniformly continuous. Prove that [itex]f|(a,b)→ℝ[/itex] is bounded.

2. Relevant equations

A function [itex]f|D→ℝ[/itex] is uniformly continuous provided that whenever {u_{n}} and {v_{n}} are sequences in D such that lim (n→∞) [u_{n}-v_{n}] = 0, then lim (n→∞) [f(u_{n}) - f(v_{n})] = 0.

A function [itex]f|D→ℝ[/itex] is bounded if there exists a real number M such that |f(x)| ≤ M for all x in D

Every bounded sequence has a convergent subsequence.

3. The attempt at a solution

[itex]f|(a,b)→ℝ[/itex] is uniformly continuous. Then for all sequences u_{n}and v_{n}in (a,b) such that lim (n→∞) [u_{n}- v_{n}] we have lim (n→∞) [f(u_{n}) - f(v_{n})] = 0.

Suppose that f is not bounded. Then for all real numbers M, there exists a number x in (a,b) such that |f(x)|> M. Further, for all natural numbers n, there exists an x_{n}in (a,b) such that |f(x_{n})|> n . Then {x_{n}} is a sequence in the bounded, open interval (a,b). Thus {x_{n}} has a convergent subsequence {x_{nk}}.

Aaand...I'm not even really sure where I was heading with that. Direction would be greatly appreciated. This is a basic real analysis course, we haven't talked about metric spaces, Cauchy-continuity, or any of that stuff. So, whatever proof the text wants should use rather simple concepts. Thanks in advance for your time.

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# Show that a uniformly continuous function on a bounded, open interval is bounded

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