Denote by $I^{(0)}$ the class of finite dimensional C*-algebras and by $I^{(k)}$ the class of unital C*-algebras which are unital hereditary C*-subalgebras of C*-algebras of the form $C(X)\otimes F$ where $X$ is a $k$-dimensional finite CW complex and $F\in I^{(0)}$ .
For a unital simple C*-algebra $A$, write $TR(A)\leq k$ if for any $\epsilon>0$ ,finite set $F\subseteq A$, and $0\neq a\in A^+$, there exist a nonzero projection $p\in A$ and a C*-subalgebra $B\in I^{(k)}$ with $1_B=p$ such that: $$\begin{align} &(1)\|px-xp\|<\epsilon\text{ for all }x\in F\\&(2)\|pxp- B\|<\epsilon \text{ for all }x\in F\\&(3) 1-p\text{ is equivalent to a projection in Her}(a)\end{align} $$
It was left as an excersice that unital simple inductive limit of $A_n$ with $TR(A_n)\leq k$ has tracial topological rank no more than $k$.
Since the only C*-subalgebras I can construct from the condition are the $\varphi_n(A_n)$, where $\varphi_n:A_n\to A$ are the canonical homomorphisms, I thought I need to show that $\varphi_n(A_n)$ are in $I^{(k)}$.
Fix one $n$, $A_n$ is a hereditary subalgebra of some algebra of form $C(X)\otimes F$. By the fact that every ideal of $A_n$ is of form $A_n\cap J$ for some ideal $J$ of $C(X)\otimes F$, I see that $A_n/\ker \varphi_n$ is a hereditary subalgebra of $(C(X)\otimes F)/J$.
However, $(C(X)\otimes F)/J$ is of form $C(Y_1)\otimes M_{n_1}\oplus ...\oplus C(Y_k)\otimes M_{n_k}$, where every $Y_j$ is a closed subset of $X$. To show $A_n/\ker(\varphi_n)$ is in $I^{(k)}$, I (maybe?) need to construct a $C(X')\otimes F'$ which contains $C(Y_1)\otimes M_{n_1}\oplus ...\oplus C(Y_k)\otimes M_{n_k}$ as a hereditary subalgebra, and I have no idea how to do this.
Also I do not know how do I use the simpleness of $A$.
Any help will be appreciated.