Realization of all distances in a decomposition of the space $R^n$ into $n+1$ parts

Let the sets $A_1, A_2, \dots, A_{n+1}$ form a covering of the $n$-dimensional euclidean space $R^n$ ($n>1$). Then among these sets can be found a set $A_i$ containing, for every $d>0$, a pair of points such that the distance between them is equal to $d$.