Prove that if number of edges is greater than ${n-1\choose 2}$, where n is the number of vertices, then G is connected

Question

My Initial thought is induction on number of edges, but I'm not very good at it

Answer 1

Among all simple graphs on $n$ vertices, the complete graph $K_n$ with $\binom{n}{2}$ edges has the maximum number of edges.

If a graph is not connected, then it has at least two components.

If a graph has two components, say with $n_1$ and $n_2$ vertices, then the maximum number of edges must be $E(n_1, n_2) = \binom{n_1}{2} + \binom{n_2}{2}$.

Show that among all two component graphs with $n_1$ and $n_2$ vertices, that $E(n_1, n_2)$ is maximized when $n_1 = 1$ or $n_2 = 1$.

If a graph has more than $2$ components, the total number of edges is strictly less than any graph where an edge is added between two of these components.