(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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**