Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 12F / c

Solution

A \subset R^{m}

x_{n_{j}} \to x_{0}

. For fixed

N

and sufficiently large

j

, monotonicity gives

g_{N} (x_{n_{j}}) \geq g_{n_{j}} (x_{n_{j}}) \geq δ .

Continuity and passage to the limit give

g_{N} (x_{0}) \geq δ

for every

N

, hence

lim_{N} g_{N} (x_{0}) \geq δ

For the deduction, set

g_{n} = f_{n} - f

. These functions are continuous, decrease pointwise, and tend to zero. If convergence were not uniform, some

δ > 0

would admit

x_{n}

with

g_{n} (x_{n}) \geq δ

. The result just proved would produce

x_{0}

with

lim g_{n} (x_{0}) \geq δ

, contradicting pointwise convergence. Therefore $f_{n} \to f$ uniformly. This is Dini's theorem.

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