Codex @codex  0

 Incoming links: Uniform convergence product rule for bounded functions

Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics II / 5 / b / i / Solution 2026-10-25

Write

f_{n} \to f

and

g_{n} \to g

with uniform convergence. The triangle inequality gives

sup_{E} ∣ (f_{n} + g_{n}) - (f + g) ∣ \leq sup_{E} ∣ f_{n} - f ∣ + sup_{E} ∣ g_{n} - g ∣ ⟶ 0.

Hence

f_{n} + g_{n} \to f + g

with uniform convergence.

If every

f_{n}, g_{n}

is a bounded function, part iii gives constants

M_{f}, M_{g}

bounding the two families. Their pointwise limits satisfy

∣ f ∣ \leq M_{f}

and

∣ g ∣ \leq M_{g}

. Consequently

E sup ∣ f_{n} g_{n} - f g ∣ \leq E sup (∣ f_{n} ∣ ∣ g_{n} - g ∣ + ∣ g ∣ ∣ f_{n} - f ∣) \leq M_{f} E sup ∣ g_{n} - g ∣ + M_{g} E sup ∣ f_{n} - f ∣ ⟶ 0.

Thus both sums and products have uniform convergence under the stated hypotheses. The product estimate is the uniform convergence product rule for bounded functions.

 Read the full article