If is a statement that involves two variables and , then the statement is distinctly weaker than the statement . Why? Because where the first statement gives you, for each , some such that is true, the second gives you a single that works for every . In many contexts one says in the second case that the statement holds uniformly.

A simple example that illustrates the difference: for every finite set of natural numbers there exists a natural number such that for every . However, it is not the case that there exists a natural number such that every element of every finite set of natural numbers is less than or equal to . To put this a different way: every finite set of natural numbers is bounded above, but there is not a uniform bound.

There are many circumstances in mathematics where one has a statement of the form but what one really wants is its uniform version . In general, as the above example shows, a statement does not imply its uniform version. However, surprisingly often it is possible to make a statement uniform. Indeed, many of the most useful theorems and basic principles that one learns in an undergraduate mathematics course can be regarded as tools for doing precisely that. This article gives several examples of statements that can be made uniform, with a different tool used for each. Much more can be said about the tools themselves: this will be left to other articles.

Some of the examples below are slightly artificial in the sense that although we are trying to prove a statement of the form , one would not normally regard oneself as starting with the statement . But they are still examples of situations where one is trying to prove a uniform statement.

Before we start, let us think about the kinds of conditions that will have to be satisfied if we are to have any hope of changing into . For each , let be the set of all such that . Then what we are asking for is that the sets should have a non-empty intersection. So if, for instance, we find that two of them are disjoint then we are immediately doomed. In the example earlier, was a finite set of positive integers, and the sets all had the form . Any two of these have finite intersection, but the intersection of all of these sets is empty.

If we want the sets to have non-empty intersection, then two things will help us: we would like the sets to be large in some way, and we would like there not to be too many different . It is often easier to think of the complements of the being small, and then what we want is for the union of the not to consist of every .