For positive integers $n, k$ with $1 \le k \le n$, define $$L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1)$$Let $f(n)$ be the largest value of $k$ such that $L(n, 1) < L(n, 2) < ... < L(n, k)$. Prove that $f(n) < 3\sqrt{n}$ and $f(n) > k$ if $n > k! + k$.