Factorial bound for a Volterra iterate ($\Vert {\tau }^{n}f(t)\Vert \le \frac{C^n{t}^n}{n!}{\sup }_{s\le t}\Vert f(s)\Vert$)

If $\Vert \tau f(t)\Vert \le C{\int }_0^t\Vert f(s)\Vert ds$, repeated integration produces the time-ordered simplex of volume $t^n/n!$. Thus the Neumann series ${\sum }_{n\ge 0}{\tau }^n$ converges on every finite interval, without requiring the single-step operator norm to be less than one.