Projective limit of finite dimensional C* algebras

Question

Let $A$ be a separable unital $C^*$-algebra and

$A$ = $I_0 \supset I_1 \supset I_2 \supset \ldots$

Be a sequence of ideals in $A$ such that:

1. $I_k$ is ideal in $I_m$ when $k \geq m$
2. $\bigcap I_k = \varnothing$
3. $I_m / I_k$ is finite dimensional
4. Each $I_k$ is unital(of course with different unit than in $A$)

Is it true that $A \simeq \varprojlim (A / I_k) $ ?