 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-3/12f/b/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

False. Take the uniformly continuous function

f (x) = x

. Then

g_{n} (x) - f (x)^{2} = (x + 1/ n)^{2} - x^{2} = \frac{2 x}{n} + \frac{1}{n ^{2}},

which is unbounded as a function of

x

for every fixed

n

. Thus

g_{n}

does not converge uniformly to

f^{2}

R

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