Let $(f_k)$ be a uniformly convergent sequence of bounded functions on $[0,1]$. Suppose that the sequence $(f_k)$ converges uniformly on $[0,1]$ to a function $f$. Determine if
$\lim_{n \to +\infty}\int_0^{1-\frac{1}{n}} f_n(x)\,dx=\int_0^1 f(x)\,dx.$
So far i have proved that $f$ is continuous on the interval, however am confused
I can prove that $\lim_{n \to +\infty}\int_0^{1-\frac{1}{n}} f_n(x)\,dx=\int_0^{1-\frac{1}{n}} f(x)\,dx.$
Does the fact that the interval lies in $[0,1]$ make a difference as $\frac{1}{n}$ of $0,1$ is either $0$ or undefined, and do i use this with the FTC?