*NOTE: provide a brief yet clear explanation. Prove by contradiction that there

Question

*NOTE: provide a brief yet clear explanation.
Prove by contradiction that there not exist a largest integer [ Hint: Observe that for any integer n there is a greater one, any n + 1 So begin your proof Suppose for contradiction that there is a largest integer be n,... ] What is wrong with the following proof that 1 is the largest integer? Let n be the largest integer. Then, since 1 is an integer we must have 1 n. On the other hand, since n2 is also an integer we must have n2 n from which it follows that n 1. Thus, since 1 n and n 1 we must have n = 1 Thus 1 is the largest integer as claimed. What does this argument prove? Prove by contradiction that there does not exist a smallest positive real number.

Explanation / Answer

INSTEAD OF SAYING N IS THE LARGEST INTEGER , LET US SAY ..... LET THERE BE ONE LARGEST FINITE INTEGER AND LET IT BE N... N*N<=N IS WRONG ALWAYS SINCE IF N IS ANY NUMBER >1 THEN N*N>N .. THIS PROVES 2 THINGS THAT 1. OUR ASSUMPTION THAT THERE IS ONE LARGEST FINITE INTEGER N IS WRONG 2.SINCE IF WE NAME ANY FINITE INTEGER WE CAN ALWAYS A GREATER INTEGER BY ADDING ONE TO IT...NOTE THAT N IS POSITIVE N+1>N.... 3. SO 1 IS NOT THE LARGEST INTEGER THIS PROVES THAT THERE IS NO ONE LARGEST FINITE INTEGER N ......

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