- #1
kalvin
- 4
- 0
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
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