P is contained in NP ∩ Co-NP?
A language $L$ is in $P$, we have an algorithm $A$ that runs in polynomial time that recognizes it.

It is easy to show that the complement of $L$,  $\bar{L}$, is in $P$. The algorithm to recognize it is to simulate $A$ but just invert the answer.

As long as $\bar{L}$ is in P (thus in NP), $L$ is in $co-NP$.