Let $p: E\to B$ be a covering map; let $B$ connected. Show that if $p^{-1}(b_0)$ has $k$ elements for some $b_0 \in B$, then $p^{-1}(b)$ has $k$ elements for every $b \in B$.
I know that $E$ has a unique slice because $B$ is connected, but I don't know what to do next.
For the sake of providing some context, this is Section 53, Exercise 3 of Munkres' Topology.