Inequality proof involving Infs and Sups

[homegrown898]

Homework Statement

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}

Homework Equations

None

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

 

[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.