In this chapter, we present the algorithm of approaching renewal points for two independent renewal schemes to a distance less than or equal to some H > 0 appearing as a result of alternating overjumps for renewal points of these renewal schemes. We define a renewal scheme, informally formulate and describe the coupling problem for renewal schemes, present the corresponding approaching renewal algorithm, and give explicit upper bounds for power and exponential moments of approaching times for renewal points using the method of test function. We also give explicit upper bounds for power and exponential moments of approaching times of renewal points uniform with respect to the initial distance between renewal points.