[/quote]- sup (-M) = -(-a) = a
inf M = -sup(-M)
is this good enough proof or am i missing something?[/QUOTE]
No, it isn't. The fact that you haven't actually used an properties of "lim" and "sup" themselves should have been a clue! Use the definitions of "inf" and "sup". The inf M is, first of all, a lower bound on M. That is, if , then x> inf M. Can you show that -(sup(-M)) is a lower bound? If [mathj]x\in M[/tex] then . Since sup(-M) is an upper bound, we must have -x< sup(-M). Multiply on both sides by -1: x> -sup(-M).
But you still have to prove that -sup(-M) is the greatest lowerbound. To do that let m be any lower bound on M. You will need to show that -m is then an upper bound on -M. Because "sup(-M)" is the least upper bound, you now have -m> sup(-M). Now multiply on both sides by -1.