I am using Spivak Calculus. I have a general doubt regarding the definition of least upper bound of sets.(adsbygoogle = window.adsbygoogle || []).push({});

Let A be any set of real numbers and A is not a null set. Let S be the least upper bound of A.

Then by definition "For every x belongs to A, x is lesser than or equal to S"

Let M be an upper bound such that M is less than S and M does not belong to A(Assertion 1)

Then, it leads to contradiction because A is bounded above by M and M is less than least upper bound S, which means an upper bound is less than least upper bound.

So, we have either M=S or M>S. So, Assertion 1 is false.

Then by converse of Assertion 1, if N is less than S, it may belong to A but can't be an upper bound or

if N is less than S, it may not belong to A. (Assertion 2)

Is my logic right?

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# B Doubt regarding least upper bound?

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