Prove $|d_{s}(u)-d_{s}(v)|\leq1$ in BFS
First of all your question is quite confusing:

There is at least one edge $w \in V$

Is it an edge or a vertex?
Maybe it would be easier for you to look at some edge $(u,v) \in E$ and think about the moment in time $\tau$, when the BFS algorithm reached either $u$ or $v$ for the first time (it could reach them both at the same time).