I am reading chapter 4 of Nielson and Chuang's QCQI book.
I cannot prove the inequality from (4.66) to (4.67) in page 195.
That inequality is the following:
$$ |\langle\psi|U^\dagger M|\Delta\rangle|+|\langle\Delta|MV|\psi\rangle| \leq \|{|\Delta\rangle}\| + \| |\Delta\rangle \|$$
$U,V$ are arbitrary unitary operators, $|\psi\rangle$ is an arbitrary state, $M$ is an POVM element, and $|\Delta\rangle = (U-V)|\psi\rangle$.
How can I prove this inequality?