 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ib/paper-4/3h/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

For each fixed

x

x / n \to 0

1

. Moreover

1 - cos u = 2 sin^{2} (u /2) \leq u^{2} /2

gives

sup_{0 < x < 1} ∣ cos (x / n) - 1∣ \leq \frac{1}{2 n ^{2}} ⟶ 0.

Thus the limit is $1$ , with uniform convergence on $(0, 1)$ .

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