Sorry, let me explain more. Its fungus, but its not anything "serious". It is a small tear, wound, in the tail, that gets infected. But since its cartilage, there is no point for antibiotic, as it will normally clear up on its own. Are you using marine salt? Marine salt has anti-fungal properties, but Aquarium salt absolutely does not. There are plenty of anteseptics that would kill fungus, but its not really necessary, simply increase the heat and the salinity/hardness of the water, keep it really clean, and it should go away on its own, If it dosent, treat with an antiseptic type med such as methylene blue.
If you think of how fungus works, it does not "eat" live skin or tissue with blood supply running thru it. It does not kill a tree in nature, it breaks down a DEAD log. Dead stuff. So, there is no such thing as a "fungal" infection that would be life-threatinging. It always consumes dead material, or in this case, cartilage of the fins.