Proving at least $n-1$ comparisons are needed to test if an array is sorted
You can use a simple adversary argument (in Jeff's lecture note) to prove that every correct algorithm has to compare $a_i$ with $a_{i+1}$ for $i=0,1, \ldots, n−2$. 

If it is not the case, you can carefully choose the values such that $a_{i+1} &lt; a_{i}$, without violating any other comparison results. 

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