I was doing some more reading on general equilibrium, when I came across On the Fundamental Theorems of General Equilibrium by Maskin and Roberts, which gives a succinct proof of the existence of general equilibrium, as well as the two subsidiary results known as the first and second welfare theorems. It’s particuarly good in spelling out the mathematical relationship between the different results. (Be warned, though. It’s completely unintelligible if you don’t have a separate description of the model handy, though.)
I found the paper via this post by Michael Greinecker on his weblog Yet Another Sheep.
A slightly more intelligible reference for the model can be found among the notes on mathematics and mathematical economics by Kim Border. The relevant piece is the Introduction to the Arrow–Debreu–McKenzie–Nikaidô Model.
key phrase is of course “under appropriate hypotheses”
Never mind those hypotheses are as far removed from reality as is a mirage from water