# Pointwise limit of a sequence

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.

• Please write your formula out in Mathjax. This helps other users by making the question easier to search for and in case the image hosting site discontinues.
– Jam
Dec 1 '19 at 11:46

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.