The footnote is 'correct', but for totally the wrong reason. The only reason why it is 'correct' in this case is because we are considering three functions that have a uniformly equidistant from each other. This distance happens to be less than 1 and therefore the conclusion is made.

No no no (or should I say Terry Lee as it is quite obvious it is you): Proving an upper bound is not sufficient to definitely conclude that the square root of n is the closest integer. Why do you think there exists the

Squeeze Theorem? Why can't they just prove one direction of the inequality and since it converges to something, that is therefore the solution? Take a look at the 2010 HSC versus the 2002 HSC. Why can't we just suddenly conclude, from the 2002 HSC, that the limit is pi^2/6? What is it about the 2010 HSC that makes it more valid? Why do you think at the very end of the paper, they specifically stated that you can assume the other direction of the inequality?

It is very clear that both directions are required in order for it to be 'apt to declare' that P(k) is maximised, and I don't know why you are trying to justify your claim here without, at the very least, saying that the difference is a finite value less than 1.