Prove a graph with degree $\ge \frac{n}{2}$ has diameter $\leq 2$ 
$G$ is an undirected connected graph that has an even number of nodes,
  and every node has $≥ n/2$ degree. Prove that it has diameter $≤ 2$.

I understand that if two vertices are not connected, then each of them has at least $n/2$ edges connecting to the remaining vertices, so they must have a shared neighbor which would make the diameter $2$.
However, I don't know how to prove it formally through direct proof or contradiction, etc. since I am very new to graph theory and not familiar with all the principles and theorems.

If the minimal degree, denoted by $\delta(G)$, is equal to $n/2$, and the graph is connected, then every two vertices have a common neighbor (By the pigeonhole principle).
