Show $PQ$ and $QP$ have the same eigenvalues with density of $GL_n$ There is a wonderful series of lectures on YouTube of Dr. Tadashi Tokieda on Geometry and Topology. In the fourth video in this playlist Tadashi sketches an argument for why if $P$ and $Q$ are $n$ by $n$ matrices then $PQ$ and $QP$ have the same eigenvalues using the density of invertible matrices in $M_n$, the space of $n$ by $n$ matrices.
The argument goes as thus:
(1) Let $(*)$ denote the statement "$PQ$ and $QP$ have the same eigenvalues."
(2) Note that $GL_n$ is an open dense subset of $M_n$. If $Q$ is not invertible, then let $Q_n$ be a sequence of invertible matrices converging to $Q$ (with whatever norm you like).
(3) Let $(*)_n$ denote the statement "$PQ_n$ and $Q_nP$ have the same eigenvalues." Since $Q_n$ is invertible, the statement $(*)_n$ is true for every $n$. (Tadashi has already shown the claim is true in the case that $Q$ is invertible.)
(4) Now, Tadashi claims that the statement $(*)_n$ depends continuously on $n$. Therefore, as $Q_n\to Q$, and the statement $(*)_n$is true for every $n$, the statement $(*)$ is also true.
Can someone flesh out this step (4)? How exactly does $(*)_n$ depend continuously on $n$? Many thanks.

The roots of a complex polynomial are continuous in its coefficients. Hence, if $A_n$ is a sequence of matrices converging to $A$, then the eigenvalues of $A_n$ converge to those of $A$.
