Hilbert Spaces: Question on a lemma about adjoint operators

Question

I 'm studying about Hilbert spaces this semester and the following lemma states some elementary properties on adjoint operators:

I have trouble understanding why $\;(x,A^* Ax)\le \vert \vert x \vert \vert\;\vert \vert A^* Ax \vert \vert\;$. If I had $\; \vert (x,A^* Ax) \vert \;$, then the above inequality would follow from Cauchy-Schwarz inequality. So my question is, why does this hold?

Is it true to claim that $\; \vert (x,A^* Ax) \vert \;=\;(x,A^* Ax)\;$ since $\;(x,A^* Ax)=\; {\vert \vert Ax \vert \vert}^2\;$?

Any help would be valuable. Thanks in advance!!