Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 1 / 4E / ii / Solution

The sequence is at least $1$. Fix $\epsilon >0$. By the binomial theorem, for $n\ge 2$,once $(n-1){\epsilon }^2>2$. Taking positive $n$th roots gives $1\le n^{1/n}<1+\epsilon$ for all sufficiently large $n$. This proves directly from the definition of a limit of a sequence that