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uniformly continuous

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A function f:reals->reals is said to be periodic on the reals if there exists a number p greater than zero such that f(x+p)=f(x) for all x in the reals. Prove that a continuous function on the reals is uniformly continuous on the reals.

Secondary question:
Is there a way to do this with just epsilon and deltas and no need for compactness?

I have seen some ideas for this here http://answers.yahoo.com/question/index?qid=20100728093949AAGk3On
and here http://answers.yahoo.com/question/index?qid=20070813123341AAbbKGC.

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Solution Summary

This solution examines compactness of the interval.

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