Q:

How come the matrix $I - A$ where $\|A\| < 1$ is nonsingular?

As the title says, why is the matrix $I - A,$ where $I$ is the identity and $\|A\| < 1$, nonsingular? 

A:

If $I-A$ is singular then there is a nonzero vector $v$ with $(I-A)v=0$, equivalently $Av=v$. But then we get a contradiction:
$$1 = \frac{|v|}{|v|} = \frac{|Av|}{|v|} \le |A| < 1$$

