so I need to prove the following:
Prove that $n-1$ comparisons are sometimes necessary to test whether an array with $n$ distinct elements is sorted in increasing order, for any $n \geq 1$.
The problem comes with the following hint:
Assume an algorithm exists that always correctly tests if the array is sorted using at most $n-2$ comparisons and show there must exist an input where this algorithm fails.
Base on the hint, I tried proving this via contradiction and here is my attempt:
Assume an algorithm exists that correctly tests if the array is sorted using at most $n-2$ comparisons. Then, let A be an array of 2 random numbers $a$ and $b$. Given that we don't know the value of $a$ with relation to $b$ or vice versa, in order to find this relation, an thus find out if the array with n-distinct elements is sorted in increasing order, 1 comparison operation, namely between $a$ and $b$, is needed, but then $n-2 = 1-2 \neq 1$. Therefore we have arrived at a contradiction.
While this attempt is most likely wrong, I am somewhat familiar with proofs by contradiction, but I don't see how deriving a contradiction from my initial assumption (the one from the hint) helps prove that $n-1$ comparisons are necessary.