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so i just started learning this subject. i undestand the basics of it but can't seem to understand how to prove a curtain claim on it.

for example i received a question as homework that i would really appreaciate it if someone could help me with it, please explain it step by step:

  • $A \subset \mathbb R$ has an upper bound.
  • $B = \{ -a \mid a \in A\}$

prove that $B$ has a lower bound and that $\inf B = -\sup A$.

please help.. thank you.

k.stm
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DarkLeader
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    Explaining "it step by step" seems a bit too much to ask. Why don't you share your thoughts with us. If you show your effort, it's more likely that we are willing to help you. – Ertxiem - reinstate Monica Nov 11 '19 at 23:03
  • Hint: if x is an upper bound for A then is -x a lower bound for B? Can you then use this with x the supremum of A (why does A have a supremum?)? – user722227 Nov 11 '19 at 23:03
  • so according to the given that A has a upper bound i called it P and said that p is greater than or equal to every a in A. then i flipped the members so that -p <= -a for every a in A. according to the given that B is -a for every a in A and A is upper bound then B is lower bound but i can't prove that infB = -supA – DarkLeader Nov 11 '19 at 23:06
  • @DarkLeader I added the “proper symbols” to your post that you said you couldn’t write. Click “edit” to see how I did this and learn it. It’s called latex code. (And one pronounces it la-tech or something.) – k.stm Nov 11 '19 at 23:09
  • You should not say that B is a lower bound (or that A is an upper bound) because they are the sets you are trying to find bounds for! However, your logic is correct so you just need to pick a certain upper bound for A and show it can be made into a greatest lower bound for B (hint: if its not the greatest lower bound then there is a lower bound greater than it.). – user722227 Nov 11 '19 at 23:14
  • i understand the logic but i don't know how to write it in a mathematical way. i want to say that if supA >= any element in A and B = -a for every a in A then -supA = infB, but again, can't seem to understand the steps in between for a proper proof – DarkLeader Nov 11 '19 at 23:22

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