Uniform convergence of real part of sequence of holomorphic functions implies uniform convergence of {$f_n$}

Question

This question is part of an assignment which I am trying and I couldn't solve this particular question .

Question:Let f , $f_n$ (n=1,2,3...) be holomorphic functions on a region $\Omega$ . If Re($f_n$) converges uniformly to Re(f) then prove that $f_n$ converges uniformly to f.

I found this question : Uniform convergence of real part of holomorphic functions on compact sets

But it assumes 1 additional hypothesis .

So , can anyone please tell how can I prove this particular result.

thanks!!

Answer 1

The additional assumption in the link is absolutely necessary. Take $f_n(z)=in ,f(z)=0$ for a counter-example.