Hey, I have a topology problem here:

Let K be a compact subset in R and let f : K → R be a continuous function. Prove

that for every ε > 0 there exists Lε > 0 such that

|f (x) − f (y)| ≤ Lε |x − y| + ε , for every x, y ∈ K.

I'm thinking of using the Lipschitz condition, but not quite sure how to handle it. Could you anyone please give me a hint? Any input is appreciated!