Past exam of the analysis course of the University of Oxford / 2010 / A / 3 / b / Solution

If $|\lambda |>\Vert T\Vert$, write $T-\lambda I=-\lambda (I-T/\lambda )$. The Neumann series applies to $T/\lambda$, so $\lambda$ is outside the spectrum of a bounded operator. Thus $\sigma (T)\subseteq \{\lambda :|\lambda |\le \Vert T\Vert \}$ and $r(T)\le \Vert T\Vert$.

The polynomial spectral mapping theorem gives $\sigma (T^m)=\{{\lambda }^m:\lambda \in \sigma (T)\}$, and henceTaking $m$th roots and then the limit, whose existence is allowed, provesOnly this inequality is required; the reverse inequality is the additional content of the full spectral radius formula.