Suppose we have two real, symmetric and positive semidefinite matrices $A$ and $B$, and we know that they approximate each other well in the sense that
$$\| A - B \|_2 \le \epsilon,$$
where $\epsilon$ is positive and $\|\cdot\|_2$ is the $\ell_2$ operator norm. Is it possible to bound the difference between their leading eigenvalues, $|\lambda_1(A) - \lambda_1(B)|$, in terms of $\epsilon$?
I know it's possible to bound the angle between the leading eigenvectors of $A$ and $B$, but I have no idea how to prove a similar bound for eigenvalues.