Problem:
Prove that, for any given $\epsilon > 0$, the function $$ z \mapsto \frac{1}{z + i} + \sin(z) $$ has an infinite number of zeros in the strip $|\text{Im}(z) | < \epsilon$.
Attempt: I was trying to apply Rouche's theorem to this problem. But to apply this theorem, I need to specify an open set containing a circle $C$ and its interior. Then I can compare the number of zeros of holomorphic functions $f$ and $f + g$ inside this circle. However, I'm confused because here the region is an infinite strip.
In this case, I would let $f(z) = \sin(z)$ and $g(z) = \frac{1}{z+i}$. I was looking to bound $g(z)$ by $f(z)$ somehow and then argue that since $\sin(z)$ has an infinite number of zeros, so does the stated function.
Any help is appreciated.
