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**Anonymous1** Let $\displaystyle f_n(x)=\left\{\begin{array}{cc}1,&\mbox{ if for some integer i, x can be written } \frac{i}{2^n}\\0, & \mbox{ otherwise } \end{array}\right.$

(a) What is the limit of $\displaystyle f_n(x)?$

(b) Does $\displaystyle f_n(x)$ converge uniformly on $\displaystyle I = [0,1]?$

(c) Is $\displaystyle f_n$ Riemann integrable? Is it's limit?

(d) Show that $\displaystyle f$ is Lebesgue-integrable and evaluate the integral over $\displaystyle [0,1]$

I know this is a lengthy question, but I'm new to this Lebesgue stuff, so any miniscule bit of help is valued and appreciated.