= Solution
The bounded operators on a <Banach space> form a <Banach algebra> in the <operator norm>. If $\|T\|<1$, the <Neumann series> $A=\sum_{k=0}^{\infty}T^k$ converges absolutely in that <norm>. Its <partial sums> satisfy
$$
(I-T)\sum_{k=0}^NT^k=I-T^{N+1}
=\left(\sum_{k=0}^NT^k\right)(I-T).
$$
Taking limits proves $\boxed{(I-T)^{-1}=\sum_{k\geq0}T^k}$.
If $|\lambda|>\|T\|$, factor $\lambda I-T=\lambda(I-T/\lambda)$ and apply the same argument. Thus such $\lambda$ are in the <resolvent set> and
$$
\boxed{\sigma(T)\subseteq\{\lambda:|\lambda|\leq\|T\|\}.}
$$
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