Let $f:\mathbb{R}\xrightarrow{}\mathbb{C}$ be continuous, $\varepsilon:\mathbb{R}\xrightarrow{}\mathbb{R}$ be a positive and continuous function, and $g$, $h_0$ be two entire functions.
Prove: If $\text{Re}(g)<\log(\varepsilon)$ and $|f\exp(-g)-h_0|<1$, then
$$|f-h_0\exp(g)|<\varepsilon.$$
My attempt:
Multiply by $|\exp(g)|$ on both sides to get
$$|\exp(g)|\cdot|f\exp(-g)-h_0|<|\exp(g)|$$ or $$|f-h_0\exp(g)|<|\exp(g)|$$
I don't know how to use $\exp(\text{Re}(g))<\varepsilon$. If I had $|\exp(g)|<\varepsilon$ this would be immediate, but what do I do with only the real part?
If anyone is interested, I'm reading this paper by Rudin and trying to complete the last proof:
Fort any complex number $z$ we have $|e^{z}|=e^{Re(z)}$. Hence $|e^{g}|=e^{Re(g)}<e^{log(\epsilon)}=\epsilon$.