Paper 1, Section I, E

Analysis I | Part IA, 2019

State the Bolzano-Weierstrass theorem.

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

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

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

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

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