While going through F. Beukers proof of irrationality of $\zeta(3)$ I found the inequality $d_{n} < 3^{n}$ for all sufficiently large values of $n$ where $d_{n}$ denotes the LCM of all the numbers $1, 2, 3, \ldots, n$
Now this is easily established (almost obvious) once one assumes the prime number theorem. In fact we can replace $3^{n}$ by $a^{n}$ where $a > e$.
I would like to know if there is any elementary/direct proof of $d_{n} < 3^{n}$ without the recourse to the difficult prime number theorem.
An elementary proof of that $d_n=\mbox{lcm}(1,2,3,\dots,n)<3^n$ can be found in this article by D. Hanson: On the product of the primes, Canad. Math. Bull. 15(1972), 33-37. The same paper on researchgate.