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Completeness of the smooth-function metric

If

(f_{n})

is Cauchy in the smooth-function metric, each derivative sequence

(f_{n}^{(r)})

converges uniformly to a continuous function

g_{r}

. Passing to the limit in

f_{n}^{(r)} (x) - f_{n}^{(r)} (0) = \int_{0}^{x} f_{n}^{(r + 1)} (t) d t

shows that

g_{r}^{'} = g_{r + 1}

. Hence

g_{0}

is smooth and

f_{n} \to g_{0}

in the metric.

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