A heating system uses thermal energy from a reservoir at temperature T₊ to run a heat engine that in turn powers a mechanical heat pump that takes thermal energy from an environment at temperature T₋ and delivers it for space heating at temperature T. Define a suitable 1st law efficiency for this system. First assume that the heat engine dumps its waste heat into the environment at T₋. Show that the limit on the 1st law efficiency of this system is the product of the Carnot limits on the efficiency of the heat engine and the CoP of the mechanical heat pump. Next, assume that the heat engine dump sits waste heat into the interior space at temperature T. Compute the limit on the 1st law efficiency of the system and show that it is not the product of the Carnot limits on the efficiency of the heat engine and the CoP of the mechanical heat pump.