Why can't the error in proportional control system be closed by increasing the gain?
Proportional control results in a stable but non zero error when used to control a system which needs continuous input.  The classic example of this is an oven, where P control is used to control the power to a heating element. If the power is turned off, the oven cools down. Since the power commanded by the P controller is proportional to the error, then if there is constant, non-zero power then there must also be a constant, non-zero error. A PI (or PID) controller does not have this problem.

In a system where continuous input is not required, such as positioning a servo driven by a worm drive, then in theory a P controller will eventually decay towards zero error.  In a real system, trying to build such a P controller normally results in the system oscillating or "hunting" around the target, due to imperfections in the servo, sensor or control loop. To prevent that from happening, we instead design in a deadband, which is a band near zero error which is considered "good enough". Once in that band we stop trying to reduce the error.  This deadband can be arbitrarily narrow, if you're willing to pay for arbitrarily accurate sensors etc.