Prove that if \(a\) is an upper bound for \(A\), and if \(a\) is also an element of \(A\), then it must be that \(a=\sup A\)
Proof
Let \(a\) be an upper bound for \(A\) where \(a \in A\). By definition, \(\sup A\) is the least upper bound. This implies that
$$
\begin{align*}
\sup A \leq a
\end{align*}
$$
Now, since \(\sup A\) is an upper bound and since \(a \in A\), then we must also hve
$$
\begin{align*}
a \leq \sup A
\end{align*}
$$
But this implies that \(a = \sup A\) as desired. \(\ \blacksquare\)