We were shown this approach in complex analysis, but it was said this argument is not rigorous and served only to give us an intuition, so we proceeded to prove the Liouville's theorem (bounded + entire implies constant iirc) and proved FTA as a consequence.
I haven't thought about this theorem for a while, I think the idea is correct and we should easily be able to make it rigorous, I mean I didn't see any leap of faith given everything we know about continuity and convergence of sequences etc. I might be missing something, though.