Matching Upper Bound for Online Problems
Since we are dealing with deterministic algorithms, the adversary can pick the hardest input possible for the online algorithm.

Since the online algorithm we are discussing is deterministic, given some input, its output can always be determined. The adversary is 'smart' in the sense that they are able to look ahead and pick input such that it gives the worst possible result for the online algorithm.

What you have proved seems to be sufficient.