Let $(E_{n})_{n\in \Bbb{N}}$ be a sequence of finite-dimensional $C^*$-algebras such that each $E_{n}$ is a direct sum of matrix-blocks with rank at most $k$. Show that every irreducible representation of $\Pi_{n\in\Bbb{N}} E_{n}$ is of dimension$\leq k$.
I know that every irreducible representation of matrix algebra is unitarily equivalent to the identity representation. Moreover, every non-degenerate representation of $A\subseteq K(H)$ is a direct sum of irreducible representations, each of them unitarily equivalent to the identity representation.
Maybe the above observations are not-relevant...any hints or suggestions would be appreciated. Thank you.