Justification of termwise integration

Question

Please explain how the final line of the following proof implies the result of the corollary? (Note that the main middle section of the proof is for a preceding theorem.)

Answer 1

The $f_k$ are finite sums of the $g_i$, hence you have

$$\int_a^b f_k(x)\,dx = \int_a^b \sum_{i=1}^k g_i(x)\,dx = \sum_{i=1}^k \int_a^b g_i(x)\,dx$$

by the linearity of the integral. Since the series $g_i$ is supposed to converge uniformly, the sequence of the $f_k$ satisfies the premise of the theorem that limit and integral can be interchanged. Taking the limit inside the integral gives

$$\int_a^b \sum_{i=1}^\infty g_i(x)\,dx$$

and taking the limit of the sum of the integrals gives

$$\sum_{i=1}^\infty \int_a^b g_i(x)\,dx.$$