Definition of a club: Let $G$ be a graph with $n$ vertices where $n > 2$. We call the graph $G$ a club if for all pairs of distinct vertices $u$ and $v$ not connected by an edge, we have $\deg(u)+\deg(v)\ge n$. Ref: Khoussainov,B. , Khoussainova,N. (2012), Lectures on Discrete Mathematics for Computers Science, World Scientific Pg.(83)
My strategy was to prove this using the proof by cases method.
Case 1: Prove for the case when $v_0$ and $v_1$ are connected by an edge, in which case there is clearly a path from $v_0$ to $v_1$, of length $1$, so this case is proven.
Case 2: Prove for the case when $v_0$ and $v_1$ are not connected by an edge. Now, for this case, since the hypothesis is assumed to be true and Graph $G$ is a club,
$$\deg(v_0) + \deg(v_1)\ge n\;.$$
That's all I've got so far, and I'm not sure how to proceed.