Your trainer is 100% correct. When you are in the gym working out, you are actually breaking down your muscles. This causes a stress response in the body and the body rebuilds the muscle larger and stronger.
However, the recovery process takes time. If you are training the same muscle every single day, you are never allowing your body to recover and you are basically shooting yourself in the foot. You always want to wait at least 48 hours to train a muscle again.
In my opinion, there is really no reason to train a muscle more then 2-3 times per week. Hit it hard, stimulate it to grow, and then let it rest. Look at the best bodybuilders of all time - jay cutler, ronnie coleman etc - they all trained their arms only 1-2 times per week.
KISS. Keep it simple stupid.