CMO 1979 Problem 3

Let $a$ , $b$ , $c$ , $d$ , $e$ be integers such that $1 \le a < b < c < d < e$ . Prove that $$\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},$$ where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g. $[4,6] = 12$ ).