Let ${A_n}$ be a sequence of $p \times p$ symmetric positive semi-definite matrix and $A_n$ converges to a matrix $A$, that is every elements of $A_n$ converges to corresponding element of $A$.
Then can I say anything about the relationship between the eigenvaues of $A_n$ and $A$? For example, If the norm of $A_n-A$ is small enough, can I make the eigenvalues of $A_n$ close enough to that of $A$?
Since you can see $A_n$ as pertubated matrices to the exact $A$, that is diagonalizable, you can use the Bauer Fike Theorem.
It states, that the difference in eigenvalues is bounded by the condition of your eigenvector-basis and the norm of your pertubation.
Since your matrix is normal, you can set this condition number to 1.