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If there is a closed interval I = [a,b] then is it appropriate to assume that b = supI and a = infI ? Does it need to be shown? because I'm not sure it is written explicitly anywhere.
Prof. Alekseenko: It is actually a theorem that $ b=\sup{(a,b)} $ and $ a=\inf{(a,b)} $. You may assume that this theorem is given, however, it seems unusual that you need a statement like that. Intervals are rather simple. Do we have to use $ \sup $ and $ \inf $ on them?
I don't think we need to say sup and inf, because they are the end points and nothing in the interval can be larger or smaller than b and a, respectively.
Prof. Alekseenko: I agree. As a rule of thumb, $ \sup $ and $ \inf $ are used when you are not given the particular points bounding the set below and above. So that we have to use the fact that $ \sup $ and $ \inf $ for the set exist and that the $ \sup $ and $ \inf $ define the boundary points. When the boundary points of the set are given to us, for example, the limits of the interval $ (a,b)\, $ usually, we do not need to "re-invent" them.
