Eigenvalues of converging matrices

Question

Suppose we have two sequences of $k\times k$ symmetric matrices $\{A_n\}$ and $\{B_n\}$ s.t. $$ A_n-B_n\to 0 $$ (elementwise). In addition, the eigenvalues of $B_n$ are uniformly bounded from below by $c>0$. Is it true that $$ \liminf_{n\to\infty}\lambda_{min}(A_n)>0, $$ where $\lambda_{min}(A)$ is the smallest eigenvalue of $A$. I believe it is true because $$ \lambda_{min}(A_n)=\min_{x:\lVert x\rVert=1}x'A_nx\ge \min_{x:\lVert x\rVert=1}x'B_nx+\min_{x:\lVert x\rVert=1}x'(A_n-B_n)x $$ and the second term vanishes as $n\to\infty$.