Dirac's Theorem

Theorem
If a connected graph $$G$$ has $$n \ge 3$$ vertices and the degree of each vertex is at least $$\frac{n}{2}$$, then $$G$$ is Hamiltonian.

Proof
Take any two non-adjacent vertices $$u, v \in G$$.

Then $$\deg u + \deg v \ge \frac n 2 + \frac n 2 = n$$.

The result follows by a direct application of Ore's Theorem.