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I Supremum and Infimum Proof

  1. Apr 10, 2016 #1
    Let S and T be subsets of R such that s < t for each s ∈ S and each t ∈ T. Prove carefully that sup S ≤ inf T.


    I start by using the definition for supremum and infinum, and let sup(S)= a and inf(T)= b

    i know that a> s and b< t for all s and t. How do i continue? , do i prove it directly starting from s< t or will it be easier to use proof by contradiction?
  2. jcsd
  3. Apr 10, 2016 #2


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    Try contradiction.
  4. Apr 11, 2016 #3


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    Usually the definition of upper/lower bound would only imply [itex]s \leq \sup(S)[/itex] for all [itex]s \in S[/itex] and [itex]\inf(T) \leq t[/itex] for all [itex]t \in T[/itex]. In other words, the upper and lower bounds can be in the set themselves. The stated result should hold regardless though.

    Just start with [itex]\inf(T) \lt \sup(S)[/itex] and go from there. There must be an [itex]s \in S[/itex] such that [itex]\inf(T) \lt s[/itex] ( otherwise [itex]\inf(T)[/itex] would be an upper bound of [itex]S[/itex] that's less than [itex]\sup(S)[/itex] ). But then, for similar reasons, there must be a [itex]t \in T[/itex] such that [itex]t \lt s[/itex] ( fill in the details ).
    Last edited: Apr 11, 2016
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