Solution
= Solution
The sequence is at least $1$. Fix $\varepsilon>0$. By the <binomial theorem>, for $n\geq2$,
$$
(1+\varepsilon)^n\geq\binom n2\varepsilon^2
=\frac{n(n-1)}2\varepsilon^2>n
$$
once $(n-1)\varepsilon^2>2$. Taking positive $n$th roots gives $1\leq n^{1/n}<1+\varepsilon$ for all sufficiently large $n$. This proves directly from the definition of a <limit of a sequence> that
$$
\boxed{n^{1/n}\longrightarrow1.}
$$