supremum of family of increasing linear type function

Question

If we assume $f_{i}(x)=\langle x^*,x \rangle +c \| x\|$ 's are monotone increasing functions, does taking the supremum of these functions over $i \in I$ yield monotone increasing functions? I assume with $x_{1} \leq x_{2}$ then we have $f_{i}(x_{1})\leq f_{i}(x_{2})$ for all $i \in I$, since $f_{i}$ s are continuous we can take the supremum of both sides over i then it yields the result ? Is that correct ?