Proof
Let b = sup(\(R\)). Then, for \(x \ge b,\)
\begin{equation*}
F(x) = P(X \le x) = P(X \le b) + P( b \lt X \le x) = P(X \le b) = 1
\end{equation*}
since all of the x-values in this range are in R and therefore will either sum over or integrate over all of \(R\text{.}\)