Suppose function f: D R and function g: D R , with x₀ as an accumulation point of D. Functions f and g have limits at x₀. Show that if f(x) g(x) for all x D, then the lim x x₀ f (x) lim x x₀ g(x).

I understand the limits exists for both functions, and utilized the definitions of a limit to start the proof; after defining the limit of f(x) = L and the limit of g(x) = M. I've let = (L - M) 2.

I'm stuck on the algebra of the functions, please explain the steps.