# Prove supS <= infT

1. Mar 28, 2010

### The_Iceflash

1. The problem statement, all variables and given/known data
Let S and T be non-empty subsets of R, and suppose that for all s $$\in$$ S and t $$\in$$ T, we have s $$\leq$$ t.

Prove that supS $$\leq$$ infT.

2. Relevant equations
N/A

3. The attempt at a solution

Since s $$\in$$ S $$\Rightarrow$$ s $$\in$$ T, supT is an upper bound for S.
Since supS is the least upper bound, supS $$\leq$$ supT.

How does this look to you? Feedback is appreciated.

2. Mar 28, 2010

### Office_Shredder

Staff Emeritus
$$supS \leq supT$$ doesn't seem that useful as a start to be honest (supT and infT aren't very close to each other in general). To show that $$supS \leq infT$$, can you show that infT is an upper bound of S?