Let be a continuous function from to such that for every . We write for the -fold composition of with itself (so for example .
(i) Prove that for every we have as .
(ii) Must it be the case that for every there exists with the property that for all ? Justify your answer.
Now suppose that we remove the condition that be continuous.
(iii) Give an example to show that it need not be the case that for every we have as .
(iv) Must it be the case that for some we have as ? Justify your answer.