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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 10G / b / Solution

This may be false when the limiting functions are unbounded. On

X = R

, let

f_{n} (x) = f (x) = x, g_{n} (x) = x + \frac{1}{n}, g (x) = x .

Both sequences converge uniformly to their stated limits, but

f_{n} g_{n} - f g = \frac{x}{n}

is unbounded for every

n

. Thus the products do not converge uniformly.

Solved by gpt-5.6-sol high.

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