prove that if A has the least upper bound property, then it also has the greatest lower bound property.(adsbygoogle = window.adsbygoogle || []).push({});

the least upper bound property means that if A has an upper bound, then it also has a least upper bound. the greatest lower bound property means that if A has a lower bound, then it also has a greatest lower bound..

the common proof is to simply show that the least upper bound of all the lower bounds of a subset B of A is equal to the greatest lower bound of B. i know about this proof, but i came up with my own proof (and my own lemma). can someone check if it is correct? i personally found no logical error. i typed it out in ms word (with capitilized sentences).

my idea was to first show that if < is an order relation, then > is also an order relation, then you can imagine what i did next (lack of glb would mean lack of lub, a contradiction).

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# Check my alternate proof, please

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