Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 1 / 4E (Analysis I)

Every bounded sequence in $\mathbb{R}$ has a convergent subsequence.

$a_n\to a$ means that for every $\epsilon >0$ there is $N$ such that $n\ge N$ implies $|a_n-a|<\epsilon$.

If $(a_n)$ is bounded, apply Bolzano–Weierstrass to $(\Re a_n,\Im a_n)$ (or successively to both coordinates). If it is unbounded, recursively choose $n_k>n_{k-1}$ with $|a_{n_k}|\ge k^2$. The alternatives are not exclusive: $0,1,0,4,0,9,\dots$ has both kinds of subsequence.