I am currently reading 'Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach' by J. Hubbard and B. Hubbard. In the first chapter, there is the proposition:

> 
> Let A be a square matrix. If $|A|<1$, the series $$S=I+A+A^2+\cdots$$
>  converges to $(I-A)^{-1}$.
> 
> 
> 

The proof first shows that
$$S\_k(I-A)=I-A^{k+1}$$
and similarly
$$(I-A)S\_k=I-A^{k+1}$$
where $S\_k$ is the sum of the first $k$ terms in the series. Then it shows that
$$|A^{k+1}|\leq|A|^{k+1}$$
and according to the proof in the book, it can be said from this that $\lim\_{k\to \infty}A^{k+1}=0$ when $|A|<1$. Consequently, $S(I-A)=I$ and $(I-A)S=I$. Therefore $S=(I-A)^{-1}$.

However I do not understand how it can be said that $\lim\_{k\to \infty}A^{k+1}=0$ when $|A|<1$. In addition, all similar propositions I have found on the internet state $|\lambda\_i|<1$ as the necessary and sufficient condition. How does this relate to the condition, $|A|<1$?