real analysis - Proving a limit using precise definition

Make a conjecture about $\lim_{n \to \infty} s_k$ , and prove your conjecture.

$s_k=
\begin{cases}
k, & \text{if $k$ is even} \\[2ex]
\frac{1}{k}, & \text{if $k$ is odd}
\end{cases}$

I have come to the conclusion that the limit does not exist? But I am unsure how to prove this using the precise definition of the limit. Do I choose a concrete number for epsilon and show that the limit will exceed that?