Limit of Integer to Reciprocal Power/Proof 2

Theorem
Let $\left \langle {x_n} \right \rangle$ be the sequence in $\R$ defined as $x_n = n^{1/n}$.

Then $\left \langle {x_n} \right \rangle$ converges with a limit of $1$.

Proof
We have the definition of the power to a real number:
 * $\displaystyle n^{1/n} = \exp \left({\frac 1 n \ln n}\right)$.

From Powers Drown Logarithms, we have that:
 * $\displaystyle \lim_{n \to \infty} \frac 1 n \ln n = 0$

Hence:
 * $\displaystyle \lim_{n \to \infty} n^{1/n} = \exp 0 = 1$

and the result.