Basic Complex Analysis: Uniform convergence of derivatives to 0

[snipez90]

Homework Statement

Let $f_n$ be a sequence of holomorphic functions such that $f_n$ converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that $f '_n$ converges to zero uniformly in D = {z : |z| < 1/2}.

Homework Equations

Cauchy inequalities (estimates from the Cauchy integral formula)

The Attempt at a Solution

Okay I am less sure about this one, but

Given $\varepsilon > 0,$ there exists N > 0 such that n > N implies
$$||f_n|| < \varepsilon.$$
Fix 0 < r < 1/2. Then for any z in D, the Cauchy inequalities imply
$$f'_n(z) \leq \frac{\sup_{z \in D} |f_n (z)|}{r} < 2\varepsilon,$$
whence we have sup norm convergence of the $f'_n$ to 0, as desired.

 

[Dick]

That looks pretty straightforward to me. I don't see anything wrong with it.