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

I'm trying to prove that if if is uniformly continuous on the inteval then it is bounded on the interval.

I thought to prove it by contradiction - assuming that there exists a uniformly continuous function on that is not bounded, and see that that leads to a contradiction.

Basically if isn't bounded, then there must be a point in such that at = 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 is uniformly continuous.

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

Thanks in advance.