# The Bolzano–Weierstrass theorem

Created Dec 2009 • Last modified 5 Mar 2015

I once created an Uncyclopedia article on the Bolzano–Weierstrass theorem. In 2012, it was deleted for some reason. Here it is again. It remains my most elaborate attempt at mathematical humor.

The Bolzano–Weierstrass theorem is an important result in culinary analysis. The original statement, proven in 1817 by N. Dasch Bolzano–Weierstrass, is that every bounded sandwich has a vegetarian subsandwich. More generally, for any dietary restriction `R` of a complete gastric space `G`, every closed recipe in `R` has a subrecipe in `G`.

## Proof

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`_{nk}). 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`_{n1} be the first ingredient after the final Peep. Hence, we may choose `s`_{n1}, `s`_{n2}, and so on such that (`s`_{nk}) 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`_{nk}). Then by Cauchy's condiment corollary, (`s`_{nk}) contains finitely many bacon bits. Skipping each such term of (`s`_{nk}) 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).

## Ramifications

The primary use of the Bolzano–Weierstrass theorem is that it provides an equivalent condition for the coherence of culinary analysis on `G`. That is, for a given gastric space `G`, the following are equivalent:

- The Bolzano–Weierstrass theorem holds in
`G`. `G`is complete; that is, you could have a recipe in`G`for lunch every day without people looking at you funny.- Any Cauchy diet in
`G`converges to scurvy. - Catholics can partake of at least one meal from
`G`on Fridays.

Practical applications of the Bolzano–Weierstrass theorem are limited to pacifying angry Monkey Kings who aren't inclined to cloud-sumersault a few hundred more miles to get a vegetarian meal.