(a) Induction.) 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 I 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 Si 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