If $A \subset B, a_0 \in A$ and $\beta$ is an upper bound of B then $\sup A \le \beta.$
Since $a_0 \in A,$ then $A \ne \emptyset,$ thus $\sup A$ exists. Since $A \subset B,$ then $\sup B$ is an upper bound of $A$. By definition, $\sup A$ is the least upper bound of $A.$ This tells us that $\sup A \le \sup B.$ Also, since $\beta$ is an upper bound of $B,$ then $\sup B \le \beta.$ This implies $\sup A \le \sup B \le \beta,$ hence $\sup A \le \beta.$
Please tell me if this is a correct way in proving such a statement. Any constructive criticism would be much appreciated. Thank you very much.