For every $w \in \mathbb{C}$ there exists a sequence {$z_n$} in $\mathbb{C}$\ {0} such that $z_n \rightarrow 0$ and $e^{1/z_n}\rightarrow w$ I need to show that for every $w \in \mathbb{C}$ there exists a sequence {$z_n$} in $\mathbb{C}$\
{0} such that $z_n \rightarrow 0$ and $e^{1/z_n}\rightarrow w$. 
 When $w=0$ I can take $z_n=-1/n$, then $\lim_ne^{1/z_n}=0$. But what should I do when $w\neq0$? Hope someone could help me out. Thanks

Use the definition of $e^z$ (or basic properties of it, depending on perspective).  It is periodic ($e^{z+2\pi i}=e^z$ for all $z$), so there are always inputs with arbitrarily large absolute value giving whatever output you want. In fact $0$ is the only value of $w$ for which you can't let $e^{1/z_n}$ be exactly $w$ for each $n$, with explicit formulas for $z_n$ being elementary, and you already solved it in case $w=0$.
