I have a doubt. In analysis we use extensively the fact the every bounded set has an infimum and a supremum.
Can we prove this?
Because for instance, when we define the Riemann Integral, we define this two objects $ I^+ = inf_\sigma S(f,\sigma) $ e $ I^- = sup\sigma S(f,\sigma) $ . Then, we say that the sets below are bounded and this imply that those quantities are well defined.
My question is: is an axiom or we can prove it?