Upper Bound of Natural Logarithm/Proof 1

Proof
From Logarithm is Strictly Concave:
 * $\ln$ is (strictly) concave.

From Mean Value of Concave Real Function:
 * $\ln x - \ln 1 \le \paren {\dfrac \d {\d x} \ln 1} \paren {x - 1}$

From Derivative of Natural Logarithm:
 * $\dfrac \d {\d x} \ln 1 = \dfrac 1 1 = 1$

So:
 * $\ln x - \ln 1 \le \paren {x - 1}$

But from Logarithm of 1 is 0:
 * $\ln 1 = 0$

Hence the result.