Real Natural Logarithm Function is Continuous/Proof 2

Proof
From Bounds of Natural Logarithm:
 * $\dfrac 1 2 < \ln \left({2}\right) < 1$

Fix $x \in \R$.

Consider $\dfrac x {\ln \left({2}\right) }$.

From Rationals are Everywhere Dense in Reals:
 * $\forall \epsilon \in \R_{>0} \exists r \in \Q : \left\vert {r - \dfrac x {\ln \left({ 2 }\right) } }\right\vert < \epsilon$

Thus:

Thus:
 * $\forall \epsilon \in \R_{>0} : \exists t \in \R_{>0} : \left\vert{ \ln\left({ t } \right) - x }\right\vert < \epsilon$

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.