Suppose S ⊂ R^n is compact, f:S-->R is continuous and f(x)>0 for every x ∈ S. Show that there is a number c>0 such that f(x)≥c for every x ∈ S.

Im assuming that c is the inf in this case, since it is bounded i know it exists (this comes from the compactness) but how can I show it is there?