Paper 1, Section I, E

State the Bolzano-Weierstrass theorem.

Let $\left(a_{n}\right)$ be a sequence of non-zero real numbers. Which of the following conditions is sufficient to ensure that $\left(1 / a_{n}\right)$ converges? Give a proof or counter-example as appropriate.

(i) $a_{n} \rightarrow \ell$ for some real number $\ell$.

(ii) $a_{n} \rightarrow \ell$ for some non-zero real number $\ell$.

(iii) $\left(a_{n}\right)$ has no convergent subsequence.

*Typos? Please submit corrections to this page on GitHub.*