 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-1/10g/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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.

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