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I have 2 statements that I need to say whether they are True or False. I do not need a proof. I would like some confirmation whether my answers are correct.

  1. Every ordered field has the least upper bound property: False. I believe that if Q is an ordered field, then since Q does not have the least upper bound property, then every ordered field cannot have the least upper bound property.

  2. Every ordered set that is bounded has a greatest lower bound. False. I think this is False due to basically the same reasoning as above. Q is an ordered set. For every subset, it can be bounded, but it does not have a greatest lower bound.

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Your conclusions are right, and the reason that you give in (1) is correct. The reason in (2) isn’t quite right, since a bounded set in $\Bbb Q$ can have a greatest lower bound: $[0,1]\cap\Bbb Q$ does, for instance. All that you can say, and it’s enough for your purposes, is that there are bounded subsets of $\Bbb Q$ with no greatest lower bound, e.g., $\{q\in\Bbb Q:|q|<\sqrt2\}$.

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