= Automatic decay of Mahler coefficients
{title2=$a_n(f)\longrightarrow0$}
Approximate a <continuous function> on $\mathbb Z_p$ uniformly by a function constant on residue classes modulo $p^r$. On this finite space, translation $S$ satisfies $S^{p^r}=I$, so $\Delta^{p^r}=(S-I)^{p^r}$ has every matrix coefficient divisible by $p$. Hence $\|\Delta^{jp^r}\|\leq p^{-j}$, and the <Mahler coefficients> of a locally constant function tend to zero. Since $|a_n(f)-a_n(g)|_p\leq\|f-g\|_\infty$, uniform approximation proves the assertion for every <continuous function>. This completes the coefficient-decay part of the <Mahler theorem>.
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