Paper 1, Section II, D

Let $\left(a_{n}\right)$ be a sequence of reals.

(i) Show that if the sequence $\left(a_{n+1}-a_{n}\right)$ is convergent then so is the sequence $\left(\frac{a_{n}}{n}\right)$.

(ii) Give an example to show the sequence $\left(\frac{a_{n}}{n}\right)$ being convergent does not imply that the sequence $\left(a_{n+1}-a_{n}\right)$ is convergent.

(iii) If $a_{n+k}-a_{n} \rightarrow 0$ as $n \rightarrow \infty$ for each positive integer $k$, does it follow that $\left(a_{n}\right)$ is convergent? Justify your answer.

(iv) If $a_{n+f(n)}-a_{n} \rightarrow 0$ as $n \rightarrow \infty$ for every function $f$ from the positive integers to the positive integers, does it follow that $\left(a_{n}\right)$ is convergent? Justify your answer.

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