Dini's theorem

In mathematics, the states (after Ulisse Dini named ) set of Dini that a monotone sequence of real-valued continuous functions with continuous boundary function converges uniformly on compacta.

Statement

Be a compact topological space,

A sequence of real-valued, continuous functions with

For all natural numbers and all and there is a continuous limit function, that is

For all, so the sequence converges uniformly to already, that is

Evidence

For a given set

Since the sequence of converges pointwise, which form a covering of which is open due to the assumed continuity. The overlap is monotone increasing, since the function sequence has this property. Because is compact, is already covered by a finite number of. Is the largest index of a finite number of coverage amounts shall be applied for all major indices. So is

From which the assertion follows.

Remark

The set of Dini is also valid for monotonically decreasing sequences, as can be seen either by a suitably adapted evidence or by transition result.