You can keep fish without filters but they do better with an established biological filter in/ on the tank.
Fin rot is caused by poor water quality and lots of rotting organic matter in the aquarium, which encourages bad bacteria and other microscopic organisms to grow and damage the fish.
As mentioned, the easiest way to treat fin rot is by doing a 75% water change and gravel cleaning the substrate every day for 2 weeks.
Make sure any new water is free of chlorine/ chloramine before it's added to the tank.
You can add some salt too if required. Add 1-2 heaped tablespoons of rock salt, sea salt, or swimming pool salt for every 20 litres (5 gallons) of tank water. Keep the salt in the tank for 2 weeks.
When you do the daily water changes, add salt to the new water before adding it to the tank so the salinity (salt level) in the tank remains stable.
After 2 weeks, stop adding salt and dilute it out by doing 20% water changes each day for a week. Then do a 75% water change 2-3 times a week from then on so the water remains clean.
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Most chemical medications are designed for aquairums with filters so you should be wary about using them on a tank without a filter.