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  "body": "\n## Proof\n\nWe apply the uniform limit theorem to the partial sums.\n\n1. 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.\n2. We show by induction on $N$ that $s_N$ is continuous. For $N = 0$, $s_0 = f_0$ is continuous.\n3. If $s_N$ is continuous, then $s_{N+1} = s_N + f_{N+1}$ is continuous, since [[sums of continuous functions are continuous|0VGMHCF4SC]].\n4. By the definition of a uniformly convergent series of functions, $s_N \\to s$ [[uniformly on $S$|08HJRTG5N7]].\n5. By steps 2, 3 and 4, the [[uniform limit theorem|0X9XT05JVC]] applies to the sequence $(s_N)$. So $s$ is continuous on $S$.\n\n## Idea\n\nEach 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.\n",
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