Let $A$ be symmetric positive definite matrix and $E$ is symmetric with $||E||_{2} < ||A^{-1}||^{-1}_{2}$ then prove that $A+E$ is symmetric positive definite.
-- \ Observation; Since $A$ is invertible and $A+E = A(I+A^{-1}E)$ since $||A^{-1}E||_{2} \leq ||A^{-1}||_{2}||E||_{2}< 1$ by assumption then $A+E$ is invertible. But I don't see the connection to show that eigenvalues of $A+E$ are positive.
Any hint would be appreciated.