Uniformly continuous and bounded

[math2006]

Let f be a real uniformly continuous function on the bounded
set A in \mathbb{R}^1. Prove that f is bounded
on A.

Since f is uniformly continuous, take \epsilon = m, \exists \delta > 0
such that
|f(x)-f(p)| < \epsilon
whenever |x-p|<\delta and x,p \in A
Now we have
|f(x)| < m + |f(p)|

Obviously i should show b + |f(p)| is bounded, but no idea how.
Could someone help me? thx

 

[AKG]

Obviously i should show b + |f(p)| is bounded, but no idea how.

What is b? And why would you want to show that b + |f(p)| is bounded? You mean bounded as a function of p? Well that's no easier than showing directly that f is bounded, so this doesn't seem to be a worthwhile approach.

The idea is simple. Fix \epsilon > 0. There exists \delta > 0 such that, well, you know. Since A is bounded, you can choose a FINITE number of points x₁, ..., x_n in A such that the intervals of radius \delta about the x_i cover A. The total variation of f on these intervals is at most 2\epsilon, so the total variation of f over all of A is at most 2n\epsilon.

Oh, and if this was homework, it should have been in the homework section.