This problem was an in-class example that became assigned as additional homework. The scratchwork was done in class, but we ran out of time to write out the formal proof.(adsbygoogle = window.adsbygoogle || []).push({});

Claim: Let E be a subset of R, E is not empty and E is bounded above. Set U = {b in R: b is an upper bound of E}. Prove that sup E = inf U.

Scratchwork:

• U is not empty because E is bounded above

• If b is an upper bound of E (i.e. b is in U), then sup E ≤ b, by the definiton of supremum. Therefore sup E is a lower bound of U.

• To show it is an infimum: if ß > (what we think is the inf) and show there is an x in U such that ß > x ≥ (what we think is the infimum)

• Let ß > sup E. Need an upper bound of E (say b*) where ß > b* ≥ sup E

• Take b* = sup E

My attempt of a writeup of a proof: Let E be a subset of R, E is not empty, and E is bounded above. U is not empty because E is bounded above. If U is the set of all upper bounds, there is an element b that is an upper bound of E (i.e. b in U). Then sup E ≤ b by the definition of supremum. Therefore, sup E is a lower bound of U. Let ß > sup E. We need an upper bound of E (say b*) where ß > b* ≥ sup E. Take b* = sup E. Then by the definition of infimum, sup E = inf U.

Is that even close to a decent proof? This class is really killing me. :|

Any help at all is greatly appreciated.

Thank you for your time.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Have scratchwork. Need to finish proof of sup E = inf U

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**