Show that if $G$ is simple a graph with $n$ vertices and the number of edges $m>\binom{n-1}{2}$, then $G$ is connected.
Solution 1:
$$ \begin{align*} r &\ge \frac{n-1}{n-r}\\ rn-r^2 &\ge n-1\\ rn-r^2-n+1 &\ge 0\\ (r-1)n-(r-1)(r+1) &\ge 0\\ (r-1)(n-r-1) &\ge 0 \end{align*} $$