A positive integer $n$ can be described as $B$-rough if all of the prime factors of $n$ strictly exceed $B$.

The first five 2-rough numbers are 1, 3, 5, 7, 9. We always include 1 by convention.

It appears to be true that the $k$th $B$-rough number will never exceed $Bk$.

Answers to this question, for example, show why it is true for finite ranges of $k$ and $B$. Is it possible to show that it is true in general without a major breakthrough in number theory?