 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/22g/a/solution

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

Let

m_{N} = in f_{n \geq N} x_{n}

and

M_{N} = sup_{n \geq N} x_{n}

. Positivity and normalization give

m_{N} \leq ϕ ((x_{n})_{n \geq N}) \leq M_{N} .

Shift invariance makes the middle term equal to

ϕ (x)

. Letting

N \to \infty

yields

lim inf x_{n} \leq ϕ (x) \leq lim sup x_{n} .

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