Upper Bound of Natural Logarithm/Proof 2

Theorem
Let $\ln y$ be the natural logarithm of $y$ where $y \in \R_{>0}$.

Then:
 * $ \ln y \le y - 1$

Proof
Let $\left\langle{ f_n }\right\rangle$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as:
 * $f_n \left({ y }\right) = n \left({ \sqrt[n]{ y } - 1 }\right)$

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

Then: