Real Natural Logarithm Function is Continuous/Proof 2

Proof
From Bounds of Natural Logarithm:
 * $\dfrac 1 2 < \map \ln 2 < 1$

Fix $x \in \R$.

Consider $\dfrac x {\map \ln 2}$.

From Rationals are Everywhere Dense in Reals:
 * $\forall \epsilon \in \R_{>0} \exists r \in \Q : \size {r - \dfrac x {\map \ln 2} } < \epsilon$

Thus:

Thus:
 * $\forall \epsilon \in \R_{>0}: \exists t \in \R_{>0}: \size {\map \ln t - x} < \epsilon$

Thus, the image of $\R_{>0}$ under $\ln$ is everywhere dense in $\R$.

From Monotone Real Function with Everywhere Dense Image is Continuous, $\ln$ is continuous on $\R_{>0}$.

Hence the result.