A valued field is complete iff its ring of intergers is complete

Question

Let $K$ be a field, and let $v$ be a valuation defined on K, and let $O$ be the ring of integers, and $M$ be the maximal ideal. How to show that $K$ is complete (i.e $K$ is equal to its completion) iff $O$ is complete. That is how to show that the ring of integers of the completion of $K$ is equal to completion of the ring of integers of $K$.