If $\mathscr{A}$ is an unital c*-algebra, $I$ is a closed ideal of $\mathscr{A}$ ,and $\mathscr{B}$ is a unital c*-subalgebra of $\mathscr{A}$ .
Show that the c*-algebra generated by $I\cup\mathscr{B}$ is $I+\mathscr{B}$.
I know that it only need to prove that $I+\mathscr{B}$ is closed, however I do not know how.
Any help would be appreciated.