How to prove P ⊆ Co-NP

My approach

Let L ∈ P

$\exists$ Turing Machine $M_1$ which decides L.

We can easily construct $M_2$ which decides $\bar{L}$

$\bar{L}$ ∈ CO-NP $\implies$ P ⊆ Co-NP

I'm not sure whether its a correct way to prove this or not. I found this method here Link

A deterministic polynomial time machine for a language $L$ can easily be converted to a non-deterministic polynomial time machine which has the same operational semantics (that is, it operates deterministically), and so it both non-deterministically and co-non-deterministically accept $L$ – just check the definitions.