Estimate on the norm of a self-adjoint operator

Question

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 !!

Answer 1

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.