Proof that a uniformly continuous function on an open interval is bounded

I'm trying to prove that if $\displaystyle f$ if is uniformly continuous on the inteval $\displaystyle (a,b)$ then it is bounded on the interval.

I thought to prove it by contradiction - assuming that there exists a uniformly continuous function $\displaystyle f$ on $\displaystyle (a,b)$ that is not bounded, and see that that leads to a contradiction.

Basically if $\displaystyle f$ isn't bounded, then there must be a point $\displaystyle c$ in $\displaystyle (a,b)$such that at $\displaystyle c$ $\displaystyle lim f$ = infinity. Therefore in any the neighborhood of the limit we can always find two values of the function which are very far away from each other, contradicting the assumption that $\displaystyle f$ is uniformly continuous.

Is this the right direction? And if it is, how do I got about formalizing it?

Thanks in advance.