---
tag: K3XJ683740
kind: proof
name: Proof of the continuity of the sum of a uniformly convergent series of functions
msc: [40A30, 26A15]
proves: 0R0MYT0BJD
uses: [0MRE6XRE5V, 0VGMHCF4SC, 08HJRTG5N7, 0X9XT05JVC]
sources:
  - url: https://www.jirka.org/ra/html/sec_liminter.html
    locator: Theorem 6.2.2
---

## Proof

We apply the uniform limit theorem to the partial sums.

1. Let $S \subseteq \R$, and let each $f_n \colon S \to \R$ be continuous. Let $s_N = \sum_{n=0}^{N} f_n$ be the partial sums of the [[series|0MRE6XRE5V]], and let $s$ be its sum.
2. We show by induction on $N$ that $s_N$ is continuous. For $N = 0$, $s_0 = f_0$ is continuous.
3. If $s_N$ is continuous, then $s_{N+1} = s_N + f_{N+1}$ is continuous, since [[sums of continuous functions are continuous|0VGMHCF4SC]].
4. By the definition of a uniformly convergent series of functions, $s_N \to s$ [[uniformly on $S$|08HJRTG5N7]].
5. By steps 2, 3 and 4, the [[uniform limit theorem|0X9XT05JVC]] applies to the sequence $(s_N)$. So $s$ is continuous on $S$.

## Idea

Each partial sum is a finite sum of continuous functions, so it is continuous. Uniform convergence of the series means uniform convergence of these partial sums. The uniform limit of continuous functions is continuous.