Effect of an added constraint to an optimization problem

Question

If you have a constrained maximization problem, and you add an additional constraint, my intuition says that since the new solution space is a subset of the original solution space, the new max objective value is upper bounded by the old max objective value. Is there a theorem that shows this?

Answer 1

If $A \subset B$, then $\sup_{x \in A} f(x) \le \sup_{x \in B} f(x)$.

If $B$ is the original feasible set, and you add a constraint, then you obtain a new feasible set $A \subset B$, so the answer to your question is yes.