Supremum and Infimum Proof

[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 ).