Basic Complex Analysis: Uniform convergence of derivatives to 0

[snipez90]

Homework Statement

Let [itex]f_n[/itex] be a sequence of holomorphic functions such that [itex]f_n[/itex] converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that [itex]f '_n[/itex] 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 [itex]\varepsilon > 0,[/itex] there exists N > 0 such that n > N implies
[tex]||f_n|| < \varepsilon.[/tex]
Fix 0 < r < 1/2. Then for any z in D, the Cauchy inequalities imply
[tex]f'_n(z) \leq \frac{\sup_{z \in D} |f_n (z)|}{r} < 2\varepsilon,[/tex]
whence we have sup norm convergence of the [itex]f'_n[/itex] to 0, as desired.

 

[Dick]

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