Help me with this, completely bounded, relatively compact, c0,...

Question

Given $c_0$ is the set of all sequences $x=(x_n)$ such that $\mathop {\lim }\limits_{n \to \infty } {x_n} = 0$. Let $||x|| = \mathop {\max }\limits_n |{x_n}|$ and $d(x,y)=||x-y||.$ Prove that $K \subset {c_0}$ is relatively compact $ \Leftrightarrow \mathop {\sup }\limits_{x \in K} ||x||$ $< $ $+\infty $ and $\mathop {\lim }\limits_{n \to \infty } \mathop {\sup }\limits_{x \in K} \mathop {\max }\limits_{n \le i < \infty } |{x_i}| = 0$.
My idea is trying to prove K is completely bounded (due to the fact that $c_0$ is complete) and by suppose K is not bounded, then I create a sequence $(x_n)$ such that $d(x_n,x_m)≥r$ for all $n,m$.