Let $A$ be a $C^*$-algebra, and let $(e_i)_{i\in I}$ be a two-sided contractive approximate identity for $A$. Under the canonical embedding of $A$ as a closed two-sided ideal in its multiplier algebra $M(A)$, the net $(e_i)_{i\in I}$ converges to $1_{M(A)}$ in the strict topology on $M(A)$.