course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Analysis II |
IB |
3 |
|
Paper 1, Section II, G |
2016 |
Let
(a) What does it mean to say that $\left(x_{n}\right){n}$ is a Cauchy sequence in $X$ ? Show that if $\left(x{n}\right)_{n}$ is a Cauchy sequence, then it converges if it contains a convergent subsequence.
(b) Let
(i) Show that for every
(ii) Show that
(iii) Let $\left(y_{n}\right){n}$ be a subsequence of $\left(x{n}\right){n}$. If $\ell, m$ are such that $y{\ell}=x_{m}$, show that
(iv) Show also that for every
(v) Deduce that $\left(x_{n}\right){n}$ has a subsequence $\left(y{n}\right)_{n}$ such that for every
and
(c) Suppose that every closed subset