Real ysis, convergence and integrals?

Let I:= [a,b] and let {ƒn} be a sequence of functions I→R that converges on I to ƒ. Suppose that each derivative ƒ'n is continuous on I and that the sequence {ƒ'n} is uniformly convergent to g on I. Prove that ƒ(x)-ƒ(a)= ∫ (a to x) g(t)dt and that ƒ'(x)=g(x) for all x∈I.