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 1 iff Definition 2
We have:

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

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

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

If $x \ne 1$, then $y \ne 0$.

Thus: