Prove that $G$ is connected if $δ(G) ≥ (v−1)/2$ Suppose $G$ is a graph has v vertices and $δ(G) ≥ (v−1)/2$. Prove that $G$ is connected.

Take any two $A$ and $B$. If they are connected we are done. Suppose they are not.
Let $N(X)$ be a set of neighbors of vertex $X$. Then $N(X)\geq (n-1)/2 $ for each vertex $X$ by assumption. Remember we have $$|N(A)\cup N(B)| = |N(A)|+|N(B)|-  |N(A)\cap N(B)|$$
If $|N(A)\cap N(B)| =0$ we have $$n-2\geq |N(A)\cup N(B)|\geq 2\cdot {n-1\over 2} =n-1$$
wich is a contradiction. So $N(A)\cap N(B) \ne \emptyset$ and thus there is some vertex which is adjacent to $A$ and $B$. 
