Lemma 36.13.2.label Let $A$ be a unital $C^{*}$-algebra, $I \subset A$ be a closed two-sided ideal, and $\angles{e_\beta}_{\beta \in B}$ be an increasing approximate identity for $I$, then for each $x \in A$,
\[\norm{x + I}_{A/I}= \inf\bracsn{\norm{x - y}_A|y \in I}= \lim_{\beta \in B}\norm{xe_\beta - x}_{A}\]
Proof, [Lemma 15.6, Zhu93]. For each $y \in I$, $ye_{\beta} \to y$. Thus
\[\limsup_{\beta \in B}\norm{xe_\beta - x}_{A} = \limsup_{\beta \in B}\norm{(x - y)(1 - e_\beta)}_{A} \le \norm{x - y}_{A}\]
As the above holds for all $y \in I$, $\limsup_{\beta \in B}\norm{xe_\beta - x}_{A} \le \norm{x + I}_{A/I}$.
On the other hand, since $I$ is a two-sided ideal, $xe_{\beta} \in I$ for all $\beta \in B$. Thus $\liminf_{\beta \in B}\norm{xe_\beta - x}_{A} \ge \norm{x + I}_{A/I}$.$\square$
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