Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 214 / 1 / a

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- Solution a

Solution

Put

α = in f_{k \geq 1} \frac{x _{k}}{k} .

Fix

k \geq 1

and write

n = q k + r

with

0 \leq r < k

. Repeated subadditivity gives

x_{n} \leq q x_{k} + x_{r},

so

\frac{x _{n}}{n} \leq \frac{q k}{n} \frac{x _{k}}{k} + \frac{x _{r}}{n} .

The finitely many values

x_{0}, \dots, x_{k - 1}

are bounded, hence

lim sup_{n \to \infty} \frac{x _{n}}{n} \leq \frac{x _{k}}{k} .

Taking the infimum over

k

gives

lim sup x_{n} / n \leq α

, while the definition of

α

gives

x_{n} / n \geq α

for every

n

. Therefore

n \to \infty lim \frac{x _{n}}{n} = k \geq 1 in f \frac{x _{k}}{k} .

This is Fekete lemma.

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