What is wrong with the following “proof” that any order for n ≥ 10 pounds of fish can be filled with only 5-pound fish?
We use the strong form of mathematical induction.
Here n0 = 10. Since an order for 10 pounds of fish can be filled with two 5-pound fish, the assertion is true for n = 10. Now let k > 10 be an integer and suppose that any order for ℓpounds of fish, 10 ≤ℓ < k, can be filled with only 5-pound fish. We must prove that an order for k pounds can be similarly filled. But by the induction hypothesis, we can fill an order for k − 5 pounds of fish, so, adding one more 5-pounder, we can fill the order for k pounds.
By the Principle of Mathematical Induction, we conclude that the assertion is true for all n ≥ 10.
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