please answer problem2 Give an example of a bounded subset o

Give an example of a bounded subset of Q which does not have a supremum; provide proof. If (F,) is an ordered field, prove that cut(F) (the field of cuts on F) is complete: every bounded set has a supremum.

Solution

Let E be a nonempty bounded subset of . E has a supremum in if and only if its supremum in is rational and that in this case, the two are equal.

Suppose consider the taylor expansion of e, from the first term keep on adding the others terms and put them in our set. All the numbers are rational, finally they converge to e which is not in Q.

Let E be a nonempty bounded subset of . E has a supremum in if and only if its supremum in is rational and that in this case, the two are equal. Suppose consider the taylor expansion of e, from the first term keep on adding the others terms and put them in our set. All the numbers are rational, finally they converge to e which is not in Q.