Couple questions of my achilles

[reefy]

I have him for 10 days and hes doing very well in my tank eating frozen and ofcourse nori. after the first 2 days it started to show ich in his body but it stop at day 8th but and today I have notice the following:
1. I notice that the upper fin has a round spot(purpleish) that I dont know what it is, please identify.
2. Why the white part of the tail is turning black? ( I notice that he lost the bottom tip of his tail) is this a sigh of curing? please say yes....