1. Consider the following blood CO₂/Ventilation system model:

            C_n+1 = C_n – bV_n +m   and   V_n+1 = aC_n   (constants a>0, b>0, m>0, 4ab<1),

Where C_n is the amount of CO₂ in the blood and V_n is the volume fraction of a breath entering the blood by this CO₂ during the nth breath. The – bV_n term represents the amount of CO₂ lost during the breath.

Prove that as n tends to infinity, C_n tends to a constant. What is that constant?