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[tex]S = \{x | x \in \mathbb{R}, x \ge 0, x^2 < c\}[/tex]

Show that c + 1 is an upper bound for S and therefore, by the Completeness Axiom, S has a least upper bound that we denote by b.

Pretty much the only tools I've got are the Field Axioms.

I think I'm supposed to do something like:

x^{2}[itex]\ge[/itex] 0. Thus c > 0.

x^{2}< c < c + 1

Thus c + 1 is an upper bound.

By the Completeness axiom, S has a least upper bound that we denote by b.

QED

It can't be just this, can it? I'm totally lost in maths, these things were dealt with ages ago and I still can't quite grasp the logic.

The part "Thus c + 1 is an upper bound" is where I think my logic fails. If this was the way (which I think is not the case) to prove c + 1 was an upper bound, couldn't we just have concluded that c is an upper bound, and thus b exists?

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# I don't get it

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