Non-existence of Equilibria in Simultaneous Auctions with a Common Budget-Constraint

I analyze an auction environment where two units of an object are sold at two simultaneous, sealed-bid, first-price auctions to bidders who have a one-dimensional type space, where a type represents the value a bidder places on each of the two units. All bidders have an identical budget constraint that binds their ability to spend in the auctions. I show that if the valuation distribution is atom-less then there does not exist any equilibrium in behavioral strategies in this auction game.