Thread: prove

inf M = -(sup (-M))

ok i tried this:

inf M = a => sup (-M) = -a
- sup (-M) = -(-a) = a
inf M = -sup(-M)

is this good enough proof or am i missing something?

Originally Posted by metlx

inf M = -(sup (-M))

ok i tried this:

inf M = a => sup (-M) = -a

This statement assumes what you are asked to prove!

[/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.

I understand what you're saying but writing it with symbols might be a problem.
We just started with axioms, lower & upper boundaries at uni.. the professor wrote on the blackboard to prove the above and none of us have any idea how.. there's also no proof in the book :/