Double Induction, etc.

15 (IMO, 1981) Let $1 \leq r \leq n$ and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest members; prove that $F(n,r) = (n+1)/(r+1)$.