They look good to me.
-Dan
Prove (i) for nonzero real numbers and integers
Prove (ii) for nonzero real numbers and integers .
Use the definition: for .
23. (i) Proof: We use induction on . Case #1: . Base case: For , . Inductive step: Suppose now as inductive hypothesis that for any real numbers and for a non-negative integers . Then . Case #2: . Base case: For , for . Suppose now as induction hypothesis that for some positive integer . Then . Conclusion: Hence, by induction, for any real numbers and for integers .
(ii) Proof: We use induction on . Case #1: . Base case: For , . Inductive step: Suppose now as inductive hypothesis that for any real numbers and for non-negative integers . Then . Case #2: . Base case: For , . Inductive step: Suppose now as inductive hypothesis that for some positive integer . Then . Case #3: . Base case: For , . (a) or (b) . Inductive step: If (a), then suppose as inductive hypothesis that . Then . Conclusion: Hence, by induction, for any real numbers and for .
For (ii) am I considering the cases correctly? Am I doing this incorrectly?
Thanks