first of all sorry to hear about your lost fish. Next IMO i would always suggest to do a fishless cycle however now you are started i would do one of two things.
1. The fastest thing to do is use tetra safestart. This stuff will instantly cycle your tank but make sure its safestart its the only product that will work.
2. keep a close eye on ammonia and nitrite levels i would suggest daily 25% water changes with Prime.
FYI i would say the cycle killed your fish and the dead fish does not hurt the cycle
1. The fastest thing to do is use tetra safestart. This stuff will instantly cycle your tank but make sure its safestart its the only product that will work.
2. keep a close eye on ammonia and nitrite levels i would suggest daily 25% water changes with Prime.
FYI i would say the cycle killed your fish and the dead fish does not hurt the cycle
