Let $L>0, F \subset C(K)$ be the set of all continous functions f with $Lip(f) \le L$ where $(K,d)$ is a compact metric space. $$Lip(f):=sup \frac{|f(x)-f(y)|}{d(x,y)}+sup|f(x)|, x \neq y, x,y \in K$$
I need to show that F is a compact subset of C(K). Per definition, this means that F has to be written as the union of finite open subcovers, right? But how could I go on from here?