The norm of a special matrix

Question

Let $A$ be hermitian matrix over the ring of quaternions with size $n\times n$ (So the eigenvalues are positive real numbers). Let $x^{*}Ax\leq |x|^{2}$ for every nonzero column vector $x$ where $|x|=\sup_{A}||Ax||$.

Then we want to prove $||A||< \infty$. In fact, we want to prove $A$ is bounded. How?