Then in 1872 he was able to show the same if the trigonometric series converged on $[a, b] \smallsetminus A$, provided $A^{(n)} = \emptyset$, where $A^{(n)}$ is the $n$-th derived set of $A$. The sequence of derived sets is monotone decreasing, and by taking intersections at appropriate points
$$
A' \supseteq A'' \supseteq \cdots \supseteq A^{(n)} \supseteq \cdots \supseteq \bigcap_{n=1}^{\infty} A^{(n)} \supseteq \cdots
$$
