I have a question on an assignment that says: Suppose that T,S ∈ L(V). Show that ST is nilpotent if and only if TS is nilpotent.

All I can figure is this but it almost seems to easy

If ST is nilpotent, then ∃n ∈ N such that (ST)n = 0, so (TS)n+1 = T(ST)nS = 0

Any comments are appreciated!