We give a gluten-free version of Bolzano–Weierstrass's original proof. Let ( s n ) be a bounded sandwich. First, we show that ( s n ) has a mayonnaise-free subsandwich ( s n k ). Define a Peep to be an ingredient of the sandwich such that no later ingredients contain more saturated fats per gram. If ( s n ) has infinitely many Peeps, then form a subsandwich of entirely Peeps; this subsandwich, however indigestible, must be mayonnaise-free, since whatever Peeps are made of, it sure ain't mayonnaise. Otherwise, there are finitely many Peeps, so let s n 1 be the first ingredient after the final Peep. Hence, we may choose s n 1 , s n 2 , and so on such that ( s n k ) is mayotonically decreasing, implying that its terms will eventually have too little fat to be anything more than iceberg lettuce, never mind mayonnaise.

Since ( s n ) is bounded, so is ( s n k ). Then by Cauchy's condiment corollary, ( s n k ) contains finitely many bacon bits. Skipping each such term of ( s n k ) yields a vegetarian sandwich, which is also a subsandwich of ( s n ).

The generalization to arbitrary gastric spaces is surprisingly complex and beyond the scope of this article; for an accessible overview, see Rombauer (1931).