SCHWEITZER 1959 Problem 3

**3.**Let $G$ be an arbitrary group, $H_1,\dots ,H_n$ some (not necessarily distinet) subgroup of $G$ and $g_1, \dots , g_n$ elements of $G$ such that each element of $G$ belongs at least to one of the right cosets $H_1 g_1, \dots , H_n g_n$ . Show that if, for any $k$ , the set-union of the cosets $H_i g_i (i=1, \dots , k-1, k+1, \dots , n)$ differs from $G$ , then every $H_k (k=1, \dots , n)$ is of finite index in $G$ . (A. 15)