# $f_n$ equicontinous on unit circle implies $u_n$ equicontinuous on closed disk

I have the following question:

Let $$f_n$$ are continuous real-valued functions on the unit circle and $$u_n$$ are harmonic functions on the open unit disk such and continuous on the closed unit disk. Furthermore, assume that $$u_n$$ agrees with $$f_n$$ on the unit circle. Show that if $$f_n$$ is equicontinuous on the unit circle, then $$u_n$$ is equicontinuous on the closed unit disk.

I think that applying the maximum principle may be useful, but I do not know how to relate that to equicontinuity. Any help would be appreciated.

Up to adding some sequence of constant functions $$C_n$$, we may assume that $$f_n$$ (hence $$u_n$$) is bounded. Thus $$f_n$$ is precompact (by Ascoli). We want to show that $$u_n$$ is precompact too, and this will conclude by the “easy” part of Ascoli.
Let $$u_{\psi(n)}$$ be a subsequence of $$u_n$$, there is a subsequence $$f_{\psi(\theta(n))}$$ that converges uniformly thus is Cauchy. Now, for any integers $$m,n$$, $$u_{\psi(\theta(m))}-u_{\psi(\theta(n))}$$ is harmonic on the open unit disk, continuous on the closed unit disk and equals $$f_{\psi(\theta(m))}-f_{\psi(\theta(n))}$$ on the boundary. So by the maximum modulus principle, $$\|u_{\psi(\theta(m))}-u_{\psi(\theta(n))}\|_{\infty} \leq \| f_{\psi(\theta(m))}-f_{\psi(\theta(n))}\|_{\infty}.$$
Thus $$u_{\psi(\theta(n))}$$ is Cauchy thus converges, which concludes.