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.