# Inequality proof involving Infs and Sups

1. Sep 25, 2011

### homegrown898

1. The problem statement, all variables and given/known data
Let F and G be bounded functions on S. If f(x) <= g(x) for all x in S prove that inf{f(x):x belongs to S} <= inf{g(x):x belongs to S}

2. Relevant equations
None

3. The attempt at a solution
Basically the idea is to let L0 = inf{f(x):x belongs to S} and L1 = inf{g(x):x belongs to S} and show that L0 <=L1

2. Sep 25, 2011

### kru_

I suggest contradiction. Assume there is some value of x where g(x) is less than the inf of all possible f(x)'s, then see if you can show why that just don't make no sense.