In Rudin's Real and Complex Analysis there's a proof of the statement that
If $z_1 ..., z_N$ are complex numbers then there is a subset $S$ of $ \{1,..., N \}$ for which $|\sum_{k \in S} z_k| \ge \frac{1}{\pi} \sum_{1}^N |z_k|$
The proof ends by stating that:
we choose $\phi _0$ such that the sum on the right:
$|\sum _{S(\phi)} z_k| = |\sum_{S(\phi) } e^{-i \phi} z_k| \ge Re(\sum_{S(\phi) } e^{-i \phi} z_k) = \sum_1^N |z_k| \cos ^+ (\alpha_k - \phi) $
is the biggest.
That maximum is at least as large as the average of the sum over $[- \pi, \pi]$ and that average is equal $\frac{1}{\pi} \sum_1^N |z_k|$, because:
$\frac{1}{2 \pi} \int_{- \pi}^{\pi} \cos^+ (\alpha - \phi) d \phi = \frac{1}{\pi}$ for every $\alpha$.
I don't know what is wrong with my understanding of this proof, because as I see it, $ \int_{- \pi}^{\pi} \cos^+ (\alpha - \phi) d \phi = 2 \cos a \le 2$ and it is supposed to be equal $2$.
(I assume that $\int_{- \pi}^{\pi} \cos^+ (\alpha - \phi) d \phi = \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \cos (\alpha - \phi) d \phi$)
Could you help me out?
It's an integral of a periodic function over an entire period. That integral is independent of which period we integrate over, so $$\int_{- \pi}^{\pi} \cos^+ (\alpha - \phi) \,d \phi=\int_{- \pi}^{\pi} \cos^+ (\phi)\, d \phi = \int_{-\pi/2}^{\pi/2}\cos(\phi)\,d\phi=\sin(\pi/2)-\sin(-\pi/2)= 2,$$ (where I also used the symmetry of the cosine function).
So it's your final assumption (in parentheses) that is wrong.