Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 1 / 9E (Analysis I)

A sequence $(a_n)$ is Cauchy when, for every $\epsilon >0$, some $N$ satisfies $|a_m-a_n|<\epsilon$ whenever $m,n\ge N$.

The general principle of convergence says that a real sequence converges exactly when it is Cauchy. Convergent sequences are Cauchy by the triangle inequality. Conversely, a Cauchy sequence is bounded, so Bolzano–Weierstrass gives a convergent subsequence $a_{n_k}\to a$; the Cauchy property then forces the entire sequence to converge to $a$.

True. For $m>n$, $|a_m-a_n|\le {\sum }_{j=n}^{m-1}{d}_j$. Convergence of $\sum d_j$ makes this tail arbitrarily small, so $(a_n)$ is Cauchy and hence convergent.

False. The sequence $a_n=(-1{)}^n/n$ converges to zero, but $|a_{n+1}-a_n|=1/n+1/(n+1)$, whose series diverges.

Writeuniformly for $j\ge 1$. The accumulated error from $j=n+1$ to $2n$ tends to zero, while the corresponding harmonic sum tends to $\log 2$. Hence $b_n\to \log 2$.