LCM of list of strictly increasing positive integers

Question

Prove that the Least Common Multiple is not less than $2019n_1$ for the following sequence of strictly increasing positive integers:

$n_1 < n_2 < \dotsb < n_{2019}$.

Answer 1

I will show that, generally, the LCM of $K$ positive integers, say $n_1<\dotsb<n_K$, is at least $K$ times the smallest of them. Indeed, denoting the LCM by $L$, we have $K$ positive integers $L/n_1,\dotsb,L/n_K$. The largest of these integers, which is $L/n_1$, must then be at least $K$; that is, $L/n_1\ge K$, as wanted.

Nice problem, but normally, you are expected to explain the source of the problem and show some work towards the solution to post.