Estimate on the norm of a self-adjoint operator EDIT: thks to Martin's comment I realize the previous version was wrong. Here is the correct version of what I need to show:
I am trying to show that if $A$ is a self - adjoint operator in a Hilbert space $H$ then 
$$
\|A\| \le \sup_{\|x\| = 1}   |\langle x, Ax \rangle|
$$
I am given the fact that whenever $\|x\| = \|y\| = 1$ we have
$$
|\langle x,Ay\rangle| \le \sup_{\|x\| = 1} \langle x,Ax \rangle.
$$ 
I am really stuck with this one, any bhint would be highly appreciated, many thanks !!

I think I have it. Take $x = Ay / \|Ay\|$, then
$$
\|Ay\| = |\frac{\|Ay\|^2}{\|Ay\|}| = |\langle x,Ay\rangle |
$$
so the inequality follows once we know that
$$
|\langle x, Ay\rangle| \le \sup_{\|x\| = 1} \langle x,Ax \rangle
$$
and the latter was given.
