> [!NOTE] Lemma (If $f_{n}$ are integrable and $f_{n}\rightrightarrows f$, then $\int f_{n} \to \int f$) > Let $(f_{n}),f_{n}:[a,b]\to \mathbb{R}$ be a sequence in [[Riemann integration|Riemann integrable functions]] that [[Uniform Convergence of Sequence of Real Functions|converges uniformly]] to $f:[a,b]\to \mathbb{R}.$ Then $f$ is Riemann integrable and $\int_{a}^{b} f_{n} \to \int_{a}^{b} f.$ **Remark**: We may write instead $\lim_{ n \to \infty }\int_{a}^b f_{n}=\int_{a}^{b} \lim_{ n \to \infty }f_{n}.$ ###### Proof To show that $f$ is Riemann integrable, using [[Riemann's criterion for integrability]], it is sufficient to show that for every $\varepsilon>0,$ there exists a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<\varepsilon.$Now, since $f_n \rightrightarrows f$ we know that for any $\varepsilon>0$ there exists $N$ such that for all $n>N,$ $\left\|f_n-f\right\|_{\infty}<\varepsilon /(4(b-a))$. For a fixed $n>N$ since $f_n$ is integrable we know that given $\varepsilon>0$ there exists a partition $P$ such that $U\left(f_n, P\right)-L\left(f_n, P\right)<\frac{\varepsilon}{2} .$Now, for that $P$ $\begin{align*} U(f, P)-L(f, P)&=\sum_{I\in P}\left[\sup _{I} f-\inf _{I} f\right]\left|I\right|\\ &=\sum_{I\in P}\left[\sup _{I}\left(f-f_n+f_n\right)-\inf _{I}\left(f-f_n+f_n\right)\right]\left|I\right|\\ &\leqslant \sum_{I\in P}\left[\left\|f-f_n\right\|_{\infty}+\sup _{I} f_n+\left\|f-f_n\right\|_{\infty}-\inf _{I} f_n\right]\left|I\right| \\ &=2 \sum_{I\in P}\lVert f-f_n \rVert_{\infty}| I |+\sum_{I\in P}\left[\sup _{I} f_n-\inf _{I} f_n\right]\ |I| \\ &\leqslant 2\left\|f-f_n\right\|_{\infty}(b-a)+U\left(f_n, P\right)-L\left(f_n, P\right) \\ &\leqslant 2 \frac{\varepsilon}{4(b-a)}(b-a)+\frac{\varepsilon}{2}=\varepsilon . \end{align*}$ To see that $\int f_n \rightarrow \int f$, notice that $\left|\int_a^b f_n-\int_a^b f\right| \leqslant\left|\int_a^b f_n-f\right| \leqslant \int_a^b\left|f_n-f\right| \leqslant \int_a^b\left\|f-f_n\right\|_{\infty}=\left\|f_n-f\right\|_{\infty}(b-a).$By the uniform convergence of $\left(f_n\right)$ to $f$, the right hand side goes to zero as $n$ goes to infinity. $\blacksquare$ # Application(s) **Corollaries**: - [[Integral of Uniformly Convergent Series of Real Functions equals Series of Their Integrals]] asserts that if partial sums $S_{n}=\sum_{k=1}^n f_{k}$ converge uniformly, then $\sum_{k=1}^\infty f_{k}$ is Riemann integrable and $\int \sum_{k=1}^\infty f_{k} = \sum_{k=1}^\infty \int f_{k}.$ - [[Uniform Convergence and Differentiability]].