Not sure if anyone here will be familiar with Walrasian equilibrium seeing as it is an econmoc conecpt although it is very much maths based. Anyhoo, I've been asked to prove the existence of a Walrasian equilibrium price vector, and I've found this:
Consider an economy with n + 1 goods X0; X1; . . . ; Xn with a price vector p = (p0; p1; . . . ; pn). Assume that an excess demand function for each good fi(p0; p1; . . . ; pn); i = 0; 1; . . . ; n, is continuous and satisfies the following condition
p0f0 + p1f1 + ... + pnfn = 0 (Walras Law):
Then, there exists an equilibrium price vector (p*0; p*1; . . . ; p*n ) which satisfies fi(p0; p1; . . . ; pn) =< 0 for all i (i = 0; 1; . . . ; n). And when pi > 0 we have fi(p*0; p*1; . . . ; p*n ) = 0.
Do you think this would suffice as an answer in an examination?