Help understanding consequence of Lemma 2.3.3 in Matrix Computations

Question

In Matrix Computations, 4th edition, I can't figure out the note at the end of Lemma 2.3.3 (page 74), the one saying

$$||(I-F)^{-1} - I||_p \le ||F||_p / (1 - ||F||_p)$$

is a consequence of the lemma.

At first glance it looks like it should follow from the applying the triangle inequality on the left hand side followed by applying the conclusion of the lemma. But that doesn't really work, as far as I can see.

So why is it a consequence?

Answer 1

Start from above(the theorem) and see that \begin{align} (I-F)^{-1} = \sum^{\infty}_{k=0}F^k \iff (I-F)^{-1}-I = \sum^{\infty}_{k=1}F^k. \end{align} Then plug this into their proof: \begin{align} \|(I-F)^{-1}-I\|_p \leq \sum^{\infty}_{k=1}\|F\|^k_p = \sum^{\infty}_{k=0}\|F\|^k_p - 1 = \frac{1}{1-\|F\|_p}-1= \frac{\|F\|_p}{1-\|F\|_p}. \end{align}