Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 2 / a / ii / Solution

Suppose $a_n\to \ell$. With tolerance $1$, choose $N$ such that $|a_n-\ell |<1$ whenever $n\ge N$. The triangle inequality gives $|a_n|\le |a_n-\ell |+|\ell |<1+|\ell |$ for these terms. SetAn empty sum is zero. This finite $B$ satisfies $|a_n|\le B$ for every $n$, so every convergent sequence is a bounded sequence.