Find the error in the following proof that all students at penn state have the same gpa. indicate the first sentence that is incorrect. explain why it is incorrect. claim: in every set of n penn state students, all students have the same gpa. proof: by induction on n. base case: n = 1. in every set containing just one penn state student, all students clearly have the same gpa. induction step: for k ≹ 1 assume that the claim holds for n = k and prove that it holds for n = k + 1. take any set s of k + 1 penn state students. remove one student from this set to obtain the set s1 with k students. by the induction hypothesis, all students in s1 have the same gpa. now replace the removed student and remove a different one to obtain the set s2. by the same argument, all students in s2 have the same gpa. therefore, all the students in s must have the same gpa.