gn:[a,b]to real R
I need to find the pointwise limit of gn and prove it converges uniformly. I managed to find its pointwise limit which is gn={0 q}. Please can anyone help with the second part. I know the definition of a pointwise function being uniformly convergent and I have no problem with the normal ones but not the ones like this. Any hint will help. Thanks.
According to your calculation of pointiwse limit function $g$ (namely, $g(x)=0$ if $x=0$ or $x$ is irrational and $g(x)=\frac 1 q$ if $x=\frac p q$ in its lowest terms) we get $g_n(x)-g(x)=\frac 1 n$ for all $x$ for all $n$. Hence $g_n \to g$ uniformly.