Let $A$ be a subhomogeneous C$^{*}$-algebra (i.e., there is a finite upper bound on the size of the irreducible representations of $A$). Let $\hat{A}$ denote its spectrum. I heard of a result that states that
if $(\pi_{n})$ is a sequence in $\hat{A}$, then $(\pi_{n})$ can converge to at most finitely many points.
I'd like to know why this statement is true. If $A$ is homogeneous, then $\hat{A}$ is Hausdorff, so the sequence can converge to at most one point in this case. I guess, in the subhomogeneous case the result has something to do with the fact that there are only finitely many choices for the dimensions of the irreducible representations. I also know that each subspace $\hat{A}_{n}:=\{\pi\in\hat{A}:\dim\pi=n\}$ is Hausdorff, but I'm not sure how to put this together. Any help is appreciated.