Does $\exp{(f_n)}$ converge to $\exp{(f)}$ uniformly if $f_n$ converges to $f$ uniformly?

Question

There is a similar question The function sequence $(\exp(f_n))$ converges uniformly if the function sequence $(f_n)$ converges uniformly. which states that if the real part of $f$ is bounded, it holds.

I was wondering if we can drop the condition that the real part of $f$ is bounded. My intuition says no, but did not come up with any reasonable counterexample or proof.

Any hints or ideas?

Answer 1

Consider $f(z)=z$ and $f_n(z)=z-\frac1n$.