I would just like to point out that i've never done a question like this before so here's my attempt:

Suppose .

Need to find such that .

Let and and fix

But:

But since .

Hence we have

Hence define

EDIT: I would really like to point out that i'm not sure if this is right. I haven't really encountered uniform continuity before!!

(I also can't see where i'm used the property that p>2....)