We present a proof of principle for the phenomenon of the Tragedy of the Commons, introduced by Lloyd in 1838, later revisited by Hardin in 1968, and currently at the center of many theories of the evolution of cooperation. The setup is a chemostat (aka bioreactor) with fluctuating operating conditions in which two species, both with the same uptake rate function and yield constant, reproduce. One of these -the cooperator- produces an enzyme required for nutrient digestion, whereas the other -the cheater- does not. We will prove that both cooperator and cheater are doomed when the cheater is present initially, establishing the tragedy for this model. On the other hand, when the cheater is absent initially, the cooperator can persist indefinitely.