CZ-SK-MO 1998 Problem 2

Given any set of $14$ (different) natural numbers, prove that for some $k$ ( $1 \le k \le 7$ ) there exist two disjoint $k$ -element subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$ such that $A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}$ and $B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}$ differ by less than $0.001$ , i.e. $|A-B| < 0.001$