Does Compactness Ensure a Positive Minimum for Continuous Functions?

[kalvin]

Let $K \subset \mathbb{R^n}$ be compact and let $f: K \rightarrow \mathbb{R}$ be continuous. Suppose that $f(x) > 0$ $\forall x \in S.$ Prove there is a $c > 0$ such that $f(x) \geq c$ $\forall x \in K$

My Sol:

I said that by the extreme value theorem $\exists a,b \in K $ such that $f(a) \leq f(x) \leq f(b) \forall x\in K$ so if we let $c=f(a) $ then $c> 0$ and $f(x) \geq c$ a number c > 0 such that f(x) ≥ c for every x ∈ K

 

[Euge]

Hi kalvin,

Assuming $S = K$, your solution is correct.