Definition of uniform continuity: such that . Apply this with to get a with . In particular, with y=0, the triangle inequality gives .
Suppose in addition that . Then .
Next (assuming from now on that ), and so . By the triangle inequality again, .
That should suggest an inductive argument: .
But multiples of the interval cover all the positive reals except for the interval . So take and , and you'll have for all real x.