(related to Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series)

- By hypothesis, $\mathbb F$ is either the field of real numbers or the field of complex numbers and $D\subset \mathbb F.$ Moreover, the infinite series of supremum norms $A:=\sum_{n=0}^\infty ||f_n||_\infty$ of functions $f_n:D\to\mathbb F$ converges to a finite limit $A < \infty.$
- By the definition of the supremum norm $|f_n(x)|\le ||f_n||_\infty$ for all $x\in D.$
- Therefore, by the direct comparison test for absolutely convergent complex series, the series $\sum_{n=1}^\infty f_n(x)$ converges absolutely for all $x\in D.$
- By definition, this means that the sequence $(\phi_n)_{n\in\mathbb N}$ of partial sums $\phi_m(x):=\sum_{n=1}^m f_n(x)$ converges uniformly to a function $f:D\to\mathbb F.$∎

**Forster Otto**: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983