Can anyone help me with the following proof please:

Let A>reals and Φ: A>reals Є C1.

Then the smallest k with

| Φ(x) – Φ(y)| ≤ |x-y| for all x,y in A is given by

k = max |Φ’(ε)| with ε ЄA.

Show that the contraction constant k is given by k = max |Φ’(ε)| with ε ЄA.

Thanks to anyone who is able to help.