I'm stuck in the following problem from Saad's Iterative Methods for Sparse Linear Systems:
In an additive projection procedure, the $k+1$th residual vector is
$$\vec{r}_{k+1} = \sum_{i=1}^p w_i (I-P_i)\vec{r}_k$$
If the weights satisfy:
$$\sum_{i=1}^p w_i = 1 \ \ \ \ \ |w_i| \leq 1 \ \ \ \ \forall i$$
and the $I-P_i$ are orthogonal projectors, show that
$$||\vec{r}_{k+1}||_2 \leq ||\vec{r}_k||_2$$.
What I tried to do was just use the triangle inequality to obtain
$$||\vec{r}_{k+1}||_2^2 \leq \sum_{i=1}^p w_i^2||(I-P_i)\vec{r}_k ||_2^2 = \sum_{i=1}^p w_i^2 (||\vec{r}_k||_2^2 - ||P_i\vec{r}_k||_2^2) \leq \sum_{i=1}^p w_i^2||\vec{r}_k||_2^2$$
Where the equality holds due to $(I-P_i)$ being an orthogonal projector.
My problem is that, from where I am now, it would be easy if the weights were a convex combination, but in this case they are an affine combination with the extra restriction that $|w_i| \leq 1$, and I don't know how to continue from here.
Thank you!