How Do Supremum and Infimum Relate When s < t for All s in S and t in T?

[wang jia le]

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.

Attempt:

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?

 

[PeroK]

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.

Attempt:

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?

Try contradiction.

 

[mbs]

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

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