As I was reading through Jitsuro Nagura's proof, I am seeing that he showed for $x \ge 2000$:
$$\psi(x) \ge 0.916x$$
where $\psi(x) = \sum\limits_{m=1}^{\infty}\vartheta(\sqrt[m]{x})$ and where $\vartheta(x) =\sum\limits_{p\le x}\log p$
Let $\text{lcm}(x) = $ the least common multiple of $\{1,2,3,\dots,x\}$.
Doesn't this imply that for $x \ge 2000$: $$\text{lcm}(x) \ge e^{0.916x} \ge (e^{0.916})^{x} > (2.499)^x$$
Am I misunderstanding Nagura's classic result?