Natural Logarithm of 2 is Greater than One Half/Proof 1

Proof
Let $f: \R_{>0} \to \R$ be the real function defined as:
 * $\forall x \in \R_{>0}: \map f x = \dfrac 1 x$

From Real Rational Function is Continuous, $\map f x$ is a continuous real function, in particular on the closed interval $\closedint a b$.

Hence the Mean Value Theorem for Integrals can be applied:

There exists some $k \in \closedint 1 2$ such that:

Thus: