In the above proof, I couldn't understand the reason given for why every boy can't rate some girl $A$ worst. First they are talking about some girl $A$ and later in the reason they are talking about each of the $n$ girls being rated worst by at least $n-1$ boys.
Can someone explain me this proof.
There are only $n$ boys, so there are $n$ total "worst" labels to distribute among the $n$ girls. That's simply not enough to give each of the $n$ girls $n-1$ or more "worst" labels. Therefore there must be at least one girl who gets fewer than $n-1$ "worst" labels.
They're not saying that it's impossible that every boy dislikes the same girl. They're saying that there must be at least one girl who doesn't get too many dislikes.