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:
 * $f_n \left({ x }\right) = n \left({ \sqrt[n]{ x } - 1 }\right)$

Let $M \left({ t }\right) = \max\left\{ { \left\vert{ t - 1 }\right\vert, \left\vert{ \dfrac{t-1}{t} }\right\vert } \right\}$

From Bounds of Natural Logarithm:
 * $ \forall t \in \R_{>0} : \left\vert{ f_n \left({ t }\right) }\right\vert \leq M \left({ t }\right)$

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

Then:

Thus: $\displaystyle \lim_{n \to \infty} f_n \left({ xy }\right) = \lim_{n \to \infty} \left({ f_n \left({ x }\right) + f_n \left({ y }\right) }\right)$ Hence the result, from the definition of $\ln$.