Hello, I am trying to prove that given a proper integral, $\displaystyle \displaystyle \int_a^b \sum_{n=0}^\infty f_n(x) dx = \sum_{n=0}^\infty \int_a^b f_n(x) dx $, assuming both converge.

Here's my attempted proof:

$\displaystyle \displaystyle \sum_{n=0}^\infty f_n(x) = \sum_{n=0}^N f_n(x) + R_N(x) $, where $\displaystyle \forall x, \; R_N(x) \to 0 $ as $\displaystyle N \to \infty $.

Thus $\displaystyle \displaystyle \int_a^b \sum_{n=0}^\infty f_n(x) dx = \int_a^b \sum_{n=0}^N f_n(x) dx + \int_a^b R_N(x) dx = \sum_{n=0}^N \int_a^b f_n(x) dx + \int_a^b R_N(x) dx $.

But $\displaystyle \displaystyle 0 \leq \left| \int_a^b R_N(x) dx \right| \leq \int_a^b \left|R_N(x)\right| dx \leq (a-b)\cdot\max_{a\leq x\leq b}\bigg\{|R_n(x)|\bigg\} \to 0 $ as $\displaystyle N \to \infty $.

Hence $\displaystyle \displaystyle \int_a^b \sum_{n=0}^\infty f_n(x) dx = \sum_{n=0}^\infty \int_a^b f_n(x) dx $.

Does this look valid?