Exercise 2, § 7

Let $\mathfrak{U}$ be an ultrafilter on $\mathbf{N}$ which is finer than the Fréchet filter, and let $X = \mathbf{N} \cup \{\omega\}$ be the space associated with $\mathfrak{U}$ (§ 6, no. 5). Show that a sequence $(x_n)$ which has an infinite number of distinct terms is never convergent in $X$.