In extreme cases the proof of an economic proposition becomes so simple that it can dispense with mathematical symbols. The first main theorem of welfare economics, according to which an equilibrium relative to a price system is a Pareto optimum, is such a case. […] In the demonstration, we study an economy consisting of a set of agents who have collectively at their disposal positive amounts of a certain number of commodities and who want to allocate these total resources among themselves. By the consumption of an agent, we mean a list of the amounts of each commodity that he consumes. And by an allocation, we mean a specification of the consumption of each agent such that the sum of all those individual consumptions equals the total resources. Following Pareto, we compare two allocations according to a unanimity principle. We say that the second allocation is collectively preferred to the first allocation if every agent prefers the consumption that he receives in the second to the consumption that he receives in the first. According to this definition, an allocation is optimal if no other allocation is collectively preferred to it. Now imagine that the agents use a price system, and consider a certain allocation. We say that each agent is in equilibrium relative to the given price system if he cannot satisfy his preferences better than he does with his allotted consumption unless he spends more than he does for that consumption. We claim that an allocation in which every agent is in equilibrium relative to a price system is optimal. Suppose, by contradiction, that there is a second allocation collectively preferred to the first. Then every agent prefers his consumption in the second allocation to his consumption in the first. Therefore the consumption of every agent in the second allocation is more expensive than his consumption in the first. Consequently the total consumption of all the agents in the second allocation is more expensive than their total consumption in the first. For both allocations, however, the total consumption equals the total resources at the disposal of the economy. Thus we asserted that the value of the total resources relative to the price system is greater than itself. A contradiction has been obtained, and the claim that the first allocation is optimal has been established.

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Debreu - “Theoretic Models: Mathematical Form & Economic Content,” 1267