I need to prove that if G is a simple graph on n vertices such that $\delta(G) \ge \lfloor n/2\rfloor$, then $G$ is connected (Where $\delta(G)$ represents the minumum degrees in $G$).
To prove this I want to show that for every vertex $u,v \in V(G)$ (where V represents the vertex set of our graph $G$) there exists a $u,v$-path in $G$.
So far I have:
Let $u,v \in V(G)$. If $u$ and $v$ are adjacent, there exists a path from $u$ to $v$ consisting of a single edge.
How do I prove this for greater lengths? Any hints would be appreciated.