Suppose S[itex]\subset[/itex]ℝn is compact, f: S-->R is continous, and f(x)>0 for every x [itex]\in[/itex]S. Show that there is a number c>0 such that f(x) ≥ c for every x[itex]\in[/itex]S. Attempt: Since S is contained in Rn is compact, then S is closed and bounded. By the extreme value thm there exists values a,b that are an absolute minimum and absolute maximum respectively. Let c = f(a). Therefore by EVT f(c) ≤ f(x) in R. Well I have the solution manual and their solution is way different to what I attempted. Is there anything right about this?