Equivalence of Definitions of Real Natural Logarithm/Proof 2

Proof
Let $x \in \R$ be such that $x > 0$.

Let $y \in \R$ be the unique number such that:
 * $e^y = x$

Definition 3 iff Definition 2
We shall show:
 * $\ds \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1} = y$

If $y = 0$, then $x = e^0 = 1$.

Thus the claim is clear, as:
 * $\forall n \in \N : n \paren {\sqrt [n] x - 1} = 0$

If $y \ne 0$, then: