Uniform Convergence Roof

Consequences of uniform convergence 10 2 proposition.

Uniform convergence roof.

Clearly uniform convergence implies pointwise convergence as an n which works uniformly for all x works for each individual x also. Suppose that f n is a sequence of functions each continuous on e and that f n f uniformly on e. If we further assume that is a metric space then uniform convergence of the to is also well defined. Uniform limit theorem suppose is a topological space is a metric space and is a.

Stack exchange network consists of 176 q a communities including stack overflow the largest most trusted online community for developers to learn share their knowledge and build their careers. Uniform convergence roof if and are topological spaces then it makes sense to talk about the continuity of the functions. In uniform convergence one is given ε 0 and must find a single n that works for that particular ε but also simultaneously uniformly for all x s. Uniform convergence means there is an overall speed of convergence.

Choose x 0 e for the moment not an end point and ε 0. Uniform convergence is the main theme of this chapter. Then f is continuous on e. Since uniform convergence is equivalent to convergence in the uniform metric we can answer this question by computing du f n f and checking if du f n f to0.

In section 2 the three theorems on exchange of pointwise limits inte gration and di erentiation which are corner stones for all later development are. In section 1 pointwise and uniform convergence of sequences of functions are discussed and examples are given. However the reverse is not true. Please note that the above inequality must hold for all x in the domain and that the integer n depends only on.

In the above example no matter which speed you consider there will be always a point far outside at which your sequence has slower speed of convergence that is it doesn t converge uniformly. We have by definition du f n f sup 0 leq x lt 1 x n 0 sup 0 leq x lt 1 x n 1. Https goo gl jq8nys how to prove uniform convergence example with f n x x 1 nx 2. The following result states that continuity is preserved by uniform convergence.

Let e be a real interval.