# Supremum

From the given we know that these exist: $\alpha = \sup \left( {f(D)} \right)\,\& \,\beta = \sup \left( {g(D)} \right)$.
$\left( {\forall z \in D} \right)\left[ {f(z) + g(z) \leqslant \alpha + \beta } \right]
This means that ${\alpha + \beta }$ is an upper bound of $\left[ {f + g} \right](D)$.