An inequality in introduction to Random Matrix theory. I am stumped in some silly easy inequality, which I don't understnad why I don't get it's on page 11 of the book by Zeitouni, et al:
http://www.wisdom.weizmann.ac.il/~zeitouni/cupbook.pdf
I am refering to the following inequality:
$$ P(\left|\left<L_N,f\right>-\left<\sigma, f\right> \right| >\delta) \le P\bigg( \left| \left<L_N,Q_\delta\right> - \left< \bar{L}_N, Q_\delta\right> \right|>\delta/4\bigg) + P\bigg( \left|\left<\bar{L}_N,Q_\delta\right> - \left<\sigma , Q_\delta\right> \right| >\delta/4\bigg) + P\bigg( \left| \left< L_N, Q_\delta \cdot 1_{|x|>B}\right>\right| > \delta/4 \bigg) $$
I don't understand why aren't there the following terms: $\left|\left<L_N,f\right>-\left<L_N,Q_\delta\right>\right| , \left| \left<\sigma , Q_\delta \right> - \left< \sigma , f\right> \right|$ , I mean if they use the triangle inequality at least it should be mentioned, even if it's zero.
I would like it if you can explain to me this inequality more clearly, at least two steps that I am missing here.
Thanks.
 A: $Q_{\delta}$ is an approximation of $f$ in certain range (in a sup norm), so he changed $f$ to $Q_{\delta}$, this comes at a cost of $\delta/8$ in the specified range, and an extra cost outside the approximation zone, which is reflected in the last summand.
The first two summands simply follow from triangle inequality + a crude union bound for the measure.
What's probably bothering you is where is the forth summand went, that's the summand responsible for the approximation of $f$ by $Q_\delta$ in the specific range, but it turns out to be okay due to the $\delta$ values he assigned to each event (there should have been an event of the form - $|Q_\delta - f|>\delta/4$ in the specified range, but due to the fact that approximation is taken in sup norm, this is an empty set and can be exempted from the union bound).
P.S. Prof. Zeitouni is a leading probabilist, and a very good teacher (from first hand experience), I would refrain from criticizing his work, especially when you're only a masters student, since the way he's written the argument is rather standardized in probability and measure theory books.
