Suppose {f_n} is a uniformly bounded sequence of holomorphic functions in D ( n=1... $). I wrote $ for infinity.

Suppose that exist such constant 0<C<$ and

|f_n(z)|<=C

for every n from N and every z from D,

and for every z from D exist limit

lim f_n(z)=f(z) for n->$.

Prove that the convergence ({f_n}->f)is uniform on every compact subset of D.

Hint: Apply the dominated convergence theorem to the Cauchy formula.

I don't know how to apply the dominated convergence theorem to the Cauchy formula.

Thank you for helping me...