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. \(\ \blacksquare\)