Nah, k=1 is the trivial base case and is allowed. The "assumption" is no such thing, but just notational convenience. One can of course tidy that up (e.g. n\leq m and trivial case is obvious), but for a random HN post it seems not worth the effort.
Well, I guess you're right. I mean, we're looking for two disctinct numbers as the final solution, but that doesn't mean we can't use k=1 in the proof.
But k cannot be equal to 1 as per your assumption that n<m. So you need to still prove a base case for the induction step.