Let $f(z)=e^{\frac{1}{z}}$, $z\in \mathbb{C}/\{0\}$, and let, for $n\in \mathbb{N}$, $$R_n=\{z=x+iy\in \mathbb{C}: |x|<\frac{1}{n}, |y|<\frac{1}{n}\}/\{0\}.$$ If for a subset $S$ of $\mathbb{C}$, $\overline{S}$ denotes the closure of $S$ in $\mathbb{C}$, then show that
$$\overline{f(R_{n})} = \overline{f(R_{n+1})}$$ Can anyone suggest how to think about this problem?
Don't let yourself be confused by the very specific definitions here. Complex analysis has some pretty strong and general theorems about the images of analytic functions which you can use. In this case, Casorati-Weierstrass is the way to go: $f$ has an essential singularity in $z=0$, so the image of any punctured neighborhood of $0$ under $f$ is dense in $\mathbb C$. Which is just shorthand for the fact that the closure is all of $\mathbb C$. Luckily, the sets $R_n$ are all punctured neighborhoods of $0$, so Casorati-Weierstrass applies.