Past exam of the mathematics course of the University of Cambridge / 2001 / ia / Paper 1 / 3D (Analysis I)

The definition of a convergent sequence isTo prove the product rule, convergence first gives an eventual bound $|u_n|\le |l|+1=:M$. ThenChoose $n$ sufficiently large that $|v_n-k|<ϵ/(2M)$ and $|u_n-l|<ϵ/[2(|k|+1)]$, as well as the bound on $u_n$. The right side is less than $ϵ$, proving $u_n{v}_n\to lk$ directly from the limit of a sequence definition.

If $l\ne 0$, eventually $|u_n-l|<|l|/2$, so the reverse triangle inequality gives $|u_n|>|l|/2$. ConsequentlyThus $1/u_n\to 1/l$. The condition $u_n\ne 0$ ensures the reciprocal sequence is defined at every index; the nonzero limit supplies the eventual uniform bound away from zero. Finally $\boxed{u_n=1/(n+1)\to 0}$ has every term nonzero, showing that this termwise property alone does not prevent a zero limit.