Second welfare theorem

Could you provide the intuition behind the second welfare theorem, and prove in a few rows the theorem itself?

Suppose you have a Pareto efficient allocation, and an equilibrium $$(x,p)$$ exists starting with this allocation as the endowment specification $$e$$. Then taking these equilibrium prices and everyone just consuming their endowment is an equilibrium; $$(e,p)$$ is an equilibrium. Indeed, since an agent's endowment is always in the budget set, and everyone chooses optimally, we must have $$x_i\succeq_i e_i$$ for every agent $$i$$. But we cannot have $$x_i\succ_i e_i$$ for some $$i$$, because then $$x$$ would be a Pareto improvement over $$e$$, which is impossible by assumption. So $$x_i\sim_i e_i$$ for every $$i$$, and just consuming the endowment is optimal given $$p$$. Since $$e$$ is also clearly feasible, we are done.
The argument is so simple because I have just assumed that the equilibrium $$(x,p)$$ exists. The standard proofs of the second welfare theorem include an existence proof, and that is also why the usual additional assumptions are needed.