Natural Logarithm of 2 is Greater than One Half

Lemma

 * $\ln 2 \ge \dfrac 1 2$

where $\ln$ denotes the natural logarithm function.

Proof of lemma
Let $f: \R_{>0} \to \R$ be the real function defined as:
 * $\forall x \in \R_{>0}: f \left({x}\right) = \dfrac 1 x$

From Rational Function is Continuous, $f \left({x}\right)$ is a continuous real function, in particular on the closed interval $\left[{a \,.\,.\, b}\right]$.

Hence the Mean Value Theorem for Integrals can be applied:

There exists some $k \in \left[{1 \,.\,.\, 2}\right]$ such that:

Thus:

Also see

 * Logarithm Tends to Infinity