On page $408$ of Nielsen & Chuang in the step going from equation $(9.48)$ to $(9.49)$, I don't see how:
$$\sum\limits_i (p_i - q_i)tr(P \sigma_i) \leq D(p_i, q_i)$$
I proceed as follows:
$$\sum\limits_i (p_i - q_i)tr(P \sigma_i)\leq \sum\limits_i |p_i - q_i| tr(P \sigma_i)$$
How is this $\leq \frac{1}{2}\sum\limits_i |p_i - q_i|$?