Suppose we have a simple C*-algebra, say $\displaystyle A$. Show that $\displaystyle M_n(A)$ is also simple.

My first guess would be to try and obtain some contradiction by assuming $\displaystyle M_n(A)$ has some closed ideal $\displaystyle J$ other than $\displaystyle 0$ or $\displaystyle M_n(A)$ and then find that $\displaystyle A$ will then have some closed ideal other than $\displaystyle 0$ or $\displaystyle A$ which will conclude the proof.

However, I'm having some trouble with the particulars...

We can suppose that there is some sequence $\displaystyle \{T^{(m)}\}\in J$ that converges to $\displaystyle T\in J$. Then we have $\displaystyle RT^{(m)},T^{(m)}R\in J$ and also $\displaystyle RT,TR\in J$ for every $\displaystyle R\in M_n(A)$.

How can we now use this to construct a closed ideal in $\displaystyle A$? (If it is at all possible)