Sum of Logarithms/Natural Logarithm/Proof 3

Proof
Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as:
 * $\map {f_n} x = n \paren {\sqrt [n] x - 1}$

Let $\map M t = \max \set {\size {t - 1}, \size {\dfrac {t - 1} t} }$

From Bounds of Natural Logarithm:
 * $\forall t \in \R_{>0}: \size {\map {f_n} t} \le \map M t$

Fix $x, y \in \R_{>0}$.

Then:

Thus:
 * $\ds \lim_{n \mathop \to \infty} \map {f_n} {x y} = \lim_{n \mathop \to \infty} \paren {\map {f_n} x + \map {f_n} y}$

Hence the result, from the definition of $\ln$.