Convergence of the integral of a series of functions

Question

I am having trouble proving that if $f_k$ converges uniformly to $f$ on $[-\pi,\pi]$ then: $$\int_{-\pi}^{\pi}f(x)dx=\lim_{k\to\infty} \int_{-\pi}^{\pi}f_k(x)dx$$

I feel like it is almost trivial, since we know that $\forall x \in [-\pi,\pi], \forall \epsilon > 0 , \exists N \in \mathbb{N}$ such that, if $k \geq N \rightarrow |f_k(x)-f(x)| \leq \epsilon $ (by uniform continuity of $f_k$).

Yet, I can´t work out a solution.

Answer 1

HINT: the key point is that $$ \left|\int_{-\pi}^{\pi }(f(x)-f_k(x))\,\mathrm d x\right|\leqslant \int_{-\pi }^{\pi }|f(x)-f_k(x)|\,\mathrm d x\tag1 $$ For $\rm(1)$ you need to know/prove that $|\int g|\leqslant \int |g|$ for any integrable $g$ (using a convergent sequence of Riemann sums and the triangle inequality the result is immediate).