KöMaL 2014 A628

Is it true that for every infinite sequence $x_1,x_2,\ldots$ of integers satisfying $|x_{k+1}-x_k|=1$ for every positive integer $k$ , there exists a sequence $k_1<k_2<\ldots<k_{2014}$ of positive integers such that as well the indices $k_1,k_2,\ldots,k_{2014}$ as the numbers $x_{k_1},x_{k_2},\ldots,x_{k_{2014}}$ (in this order) form arithmethic progressions?