Example with uniform convergence: Let denote the set of all real numbers.

In real analysis, it is the topology of uniform convergence.

The following is the more important result about uniform convergence:

This criterion for uniform convergence is often useful in real and complex analysis.

This approach justifies, for example, the notion of uniform convergence.

This type of convergence is also called almost uniform convergence.

In the absolutely summable case, the inequality proves uniform convergence.

For a continued fraction like this one the notion of uniform convergence arises quite naturally.

For functions on R this looks a lot like uniform convergence, in which case all the "hoops", once chosen, must be the same size.

Uniform convergence deals with these sorts of issues.

Uniform convergence admits a simplified definition in a hyperreal setting.

On the uniform convergence of relative frequencies of events to their probabilities.

However, one of his papers, on uniform convergence of trigonometric series, remains well cited.

This concept is often contrasted with uniform convergence.

For the Riemann integral, this can be done if uniform convergence is assumed:

Uniform convergence to a function on a given interval can be defined in terms of the uniform norm.

If the domain of the functions is a measure space then the related notion of almost uniform convergence can be defined.

In such cases, pointwise convergence and uniform convergence are two prominent examples.

It is therefore plain that uniform convergence implies pointwise convergence.

Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit.

Finally, combining all the three parts of the proof we get the Uniform Convergence Theorem.

Glivenko and Cantelli strengthened this result by proving uniform convergence of to .

It preserves uniform convergence on compact sets.

This theorem does not hold if uniform convergence is replaced by pointwise convergence.

In the above step, we were able to interchange the integral and the limit because of the uniform convergence of the integrand.