An example of a sequence of Riemann integrable functions $(f_n)$ that converges pointwise to a function $f$ that is not Riemann integrable.

real-analysis integration

For the $f_n(x)$, it comes down to the fact that the Riemann sum $1dx \rightarrow 0$ when the partition width is small-enough. Then you do this finitely-many times, and you get a total of $0$ for any $f_n(x)$. You can show that this result is true no matter how small you make the partition width $||P||$, i.e., you can show $|\Sigma f(x)dx-0 |<\epsilon$ for any partition width.

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