About the eigenvalues of real symmetric matrix

Question

Let $S$ be an $n×n$ symmetric real matrix, and for some $v≠0$, $\|Sv-αv\|<ε\|v\|$ ($α$ is a real number). Then how can we prove that $S$ has at least one eigenvalue $λ$ with $|λ-α|<ε$? I am aware that the eigenvectors associated with different eigenvalues are orthogonal, but how can I apply this? Thank you in advance.