 Factorial proof of Mertens first theorem

The prime factorization of $N!$ gives $\log N!={\sum }_{p^j\le N}\lfloor N/p^j\rfloor \log p$. The Chebyshev estimate bounds the error on replacing each floor by its argument by $O(N)$. Dividing by $N$ and using the Stirling formula gives ${\sum }_{p^j\le N}(\log p)/p^j=\log N+O(1)$. The contribution of $j\ge 2$ is bounded by the convergent series ${\sum }_{n\ge 2}(\log n)/(n(n-1))$. This proves the Mertens first theorem without the Prime number theorem.