Prove that the sequence $\displaystyle x_{n}=n^{1/n}$ converges.

I know it converges to 1. We need to use the definition of sequence convergence: $\displaystyle x_{n} \rightarrow L \Leftrightarrow(\forall\epsilon>0)(\exists N\in \mathbb{N})(\forall n\in \mathbb{N})(n>N \Rightarrow |x_{n}-L|<\epsilon)$

Another thing, obviously $\displaystyle |x_{n}-L|=|n^{1/n}-1|=n^{1/n}-1$ since $\displaystyle n^{1/n}-1\geq0$