# Limit of sequence of Riemann integrable functions is not Riemann integrable.

This post is to clarify if the following example works. Let $${q_n}$$ be an enumeration of $$\mathbb{Q}$$.

For each n define $$f_n(x)=1$$ if $$x\in\{q_1,...,q_n\}$$ and $$f_n(x)=0$$ otherwise. Then for all n $$\int f_n dx=0$$ while $$\int f dx$$ does not exist where $$lim_{n}f_n=f$$ and $$f$$ is the Dirichlet function.

Is this valid?

• yes it is correct. Notice that a countable enumeration of $\mathbb Q$ is a tautology.
– Surb
Aug 30, 2020 at 9:07
• I always thought enumeration just meant "indexed over some set" but thanks. Aug 30, 2020 at 9:10
• @JhonDoe Yes, but if "enumeration" means that each elements appears only once, then any enumeration of $\mathbb{Q}$ is countable. Aug 30, 2020 at 9:17