Basic Matrix norm question

Question

I am pretty bad at linear algebra, sorry if the question is trivial.

$P$ and $N$ are two invertible matrix .

If $||P^{-1}N||=C<1$, how can we deduce that $$||(I-P^{-1}N)^{-1}||\leq(1-C)^{-1}$$where $|\cdot|$ denote any matrix norm.

Answer 1

Write $A=P^{-1}N$, then $$(I-A)^{-1}=I+A+A^2 +A^3 +\dots$$ Hence, $$\|(I-A)^{-1}\|\le 1+\|A\|+\|A\|^2 +\|A\|^3+\ldots= 1+C+C^2 +\dots=\frac1{1-C}$$