Neumann series theorem

Question

Let $A\in \mathcal{B}(E,F)$ "the space of bounded operators from E to F (both are Banach spaces)" such that $$\|A\|<1$$ We need to show that the sequence $(S_n)_{n}$ is a Cauchy sequence where $$S_n=\sum_{k=0}^n A^k.$$ I see that if $p>q$ then $$\|S_p-S_q\| \leq \sum_{k=q+1}^p \|A\|^k$$ but how to proceed ?

Answer 1

Because the numeric series is geometric, you have $$ \sum_{k=q+1}^p\|A\|^k=\frac{\|A\|^{q+1}-\|A\|^{p+1}}{1-\|A\|}. $$ And because $\|A\|<1$, the right-hand-side can be made arbitrarily small when $p,q$ are sufficiently large. Which gives you that the sequence of partial sums is Cauchy.