A type I superconductor expels a magnetic field from its interior to a surface layer of thickness λL, the London penetration depth. λL is a function of temperature, becoming smaller as the temperature decreases. Here we analyze the process of cooling (or heating) a type I superconductor in a magnetic field, with the system remaining always in the superconducting state. We show that the conventional theory of superconductivity cannot describe this process in a consistent way. We conclude that the conventional theory of superconductivity is internally inconsistent. Instead, we suggest that the alternative theory of hole superconductivity may be able to resolve this problem.