I don't think that the point of the exercise was to be fully rigorous. Besides, I don't think I made any extraordinary leaps.
I was actually a bit euphemistic in my answer : assuming the existence of the limit in the first place and playing with it is really really not a good habit to take (no harm intended!). Especially when, like here, we are asked "if the limit exists, establish the convergence"; but certainly not only in this case.
It is not because the only "possible" limit is 1 that the sequence converges to 1.
I'm just saying...