Discuss the following “proof” of the (false) theorem:

If n is any positive integer and S is a set containing exactly n real numbers, then all the numbers in S are equal:

PROOF BY INDUCTION:

Step 1:

If , the result is evident.

Step 2: By the induction hypothesis the result is true when ; we must prove that it is correct when . Let S be any set containing exactly real numbers and denote these real numbers by . If we omit from this list, we obtain exactly k numbers ; by induction hypothesis these numbers are all equal:

.

If we omit from the list of numbers in S, we again obtain exactly k numbers ; by the induction hypothesis these numbers are all equal:

.

It follows easily that all numbers in S are equal.

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Comments, observations are welcome ðŸ™‚

Regards,

Nalin Pithwa