The class P is closed under complementation: if L is a language in P, then the complement of L is also in P. You can see this by taking any polynomial-time decider for L and switching the accept and reject states; this new machine now decides the complement of L and does so in polynomial time.
A language L is in co-NP iff its complement is in NP. So consider any language L ∈ P. The complement of L is also in P, so the complement of L is therefore in NP (because P ⊆ NP). Therefore, L is in co-NP. Consequently, P ⊆ co-NP.
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