A. Some times it is hard to do proofs in which multiple cases are needed to be proved. It is difficult seeing at first what are all the cases that need to be proved and not forgetting any.
B. The example of the students was interesting because there are time you know something exists however, by the means available it is difficult to prove. This example lead into the method of proofs that deals with proving a result based on the existence of the object possessing some specific property. Like in the example, a specific student possessing x number of hairs.
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