Defining Sequence of Natural Logarithm is Convergent

Theorem
Let $x \in \R$ be a real number such that $x > 0$.

Let $\left\langle{ x_n }\right\rangle$ be the sequence defined as:
 * $\forall n \in \N : x_n = n \left({ 1 - x^{-1/n} }\right)$

Then $\left\langle{ x_n }\right\rangle$ is convergent.