Thread: integral of a sequence of functions

1. integral of a sequence of functions

Prove that if on an interval $[a,b]$ the functions $g_{n}(x)$ are continuous and converge uniformly to $g(x),$ then on $[a,b]$

$\int_a^x g_{n}(t) dt$ converges uniformly to $\int_a^x g(t) dt.$

2. $g(x)$ is continuous and therefore integrable over $[a,b]$ 'cause $g_n\to g$ uniformly and each $g_n$ is continuous there.

Let $f_n(x)=\int_a^xg_n(t)dt,f(x)=\int_a^xg(t)dt$. So we wanna prove that $f_n\to f$ uniformly.

Let $\epsilon>0$, then $\exists N:\forall n\geq N,\left|g_n(x)-g(x)\right|<\epsilon,\forall x\in[a,b]$.
Notice that $\forall x\in[a,b]$ we have $\left|f(x)-f_n(x)\right|=\left|\int_a^xg_n(t)dt-\int_a^xg(t)dt\right|\leq \int_a^x\left|g_n(t)-g(t)\right|dt\leq \int_a^x\epsilon dt\leq \epsilon(b-a).$

3. Originally Posted by Anonymous1
Prove that if on an interval $[a,b]$ the functions $g_{n}(x)$ are continuous and converge uniformly to $g(x),$ then on $[a,b]$

$\int_a^x g_{n}(t) dt$ converges uniformly to $\int_a^x g(t) dt.$

$\forall\,\epsilon>0\,\,\exists\,N_{\epsilon}\!\!\i n\mathbb{N}\,\,s.t.\,\,\left|g_n(x)-g(x)\right|<\frac{\epsilon}{b-a}\,\,\forall\,n>N_{\epsilon}\mbox{ and }\forall\,x\in[a,b]$ , so :

$\left|\int\limits_a^x g_{n}(t) dt-\int\limits_a^x g(t) dt\right|\leq \int\limits_a^b \left|g_n(t)-g(t)\right|dt<\int\limits_a^b\frac{\epsilon}{b-a} dt=\epsilon\,,\,\,\forall\,n>N_{\epsilon}$ $\mbox{ and }\,\forall\,x\in[a,b]$.

Tonio