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How to reconcile the existence of the least upper bound?

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A set of reals that is bounded above has the least upper bound.

It is not intuitively clear to me that it should be true. I am aware of constructions of the reals from rationals where this statement is proved as a theorem, but they justify it logically and don't help my intuition. So I am looking for intuition in order to reconcile this statement to myself.

That best I could come up with is this: let $a$ be a rational number that is always greater than any element of the set and rational number $b$ is always smaller than some element of the set. Using a method like bisection search we can make the distance between $a$ and $b$ as small as we like and by that "compute" the least upper bound with any precision. I visualize the least upper bound as a decimal number, and it is clear to my intuition that If I can compute as many digits after the point as I wish for this number, this number must exist.

But the problem is that I heard of real numbers that are not computable, and if this number is in the set at hand, I will not be able to approximate the least upper bound, but the axiom that it exists still applies (I guess).


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