Given maps fi:Xi→Xi+1 for integers i<0, show that the 'reverse mapping telescope' obtained by glueing together the mapping cylinders of the fi's in the obvious way deformation retracts onto X0. Similarly, if maps fi:Xi→Xi+1 are given for all i∈Z, show that the resulting 'double mapping telescope' deformation retracts onto any of the ordinary mapping telescopes contained in it, the union of the mapping cylinders of the fi's for i greater than a given number n.