prove that if A has the least upper bound property, then it also has the greatest lower bound property. 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).