# Math Help - intotoanalysis

1. ## intotoanalysis

Let K be a bounded non-empty subset of R.Let f:K->R be a uniformly continous function.Show that f is bounded.

2. Originally Posted by sisa
Let K be a bounded non-empty subset of R.Let f:K->R be a uniformly continous function.Show that f is bounded.
Ok, so lets just establish a few things, the metric $d(p,p')$ in $\mathbb{R}^n$ is $|p-p'|$. Since assuming that $K\subset\mathbb{R}$ we have that $\phi:K\to\mathbb{R}$ is a real function. So for $\phi$ to be uniformly continous we must have that for every $\varepsilon>0$ there exists a $\delta>0$ such that $|p-p'|<\delta\implies|\phi(p)-\phi(p')|<\varepsilon$.

1. Now assume the converse and state that $\phi(p)$ is unbounded. By the fact that we are mapping to the reals let us fix $p'$ and let $\phi(p')$ be finite. Now since $\phi(p)$ is unbounded in $K$ for every $M\in\mathbb{R}$ we may find a $p\in K$ such that $M<|\phi(p)|$, so for every $M_0\in\mathbb{R}$ we can find a $p\in K$ such that $M_0<|\phi(p)-\phi(p')|$ which violates $|\phi(p)-\phi(p')|<\varepsilon$