“Almost” implies “near”

Abstract

We formulate a formal language in which it is meaningful to say that an object almost satisfies a property. We then show that any object which almost satisfies a property is near an object which exactly satisfies the property. We show how this principle can be used to prove existence theorems. We give an example showing how one may strengthen the statement to give information about the relationship between the amount by which the object fails to satisfy the property and the distance to the nearest object which satisfies the property. Examples are given concerning commuting matrices, additive sequences, Brouwer fixed points, competitive equilibria, and differential equations.