@RRL: Thanks! @YuiToCheng: I've also decided to post an answer with the correct definitions, since Ilmari only mentioned implications from limits to asymptotic statements.
@XanderHenderson The asker responded saying he/she wants to know how to fill in the "big hole" in the answer on the linked thread. It is indeed a hole, but yea Taylor series with remainder term can be made to work.
@user21820 I agree that there are some small holes in the original proof, but an answer which fills those holes would be appropriate for the original question.
Basically, the new question says "I want a better answer to an old question." Advertise in chat, offer a bounty, whatever; don't ask a new question.