Exercise 15, § 6

Let $n \to f(n)$ be a mapping of $\mathbf{N}$ onto $\mathbf{N}$ such that $f^{-1}(m)$ is finite for each $m \in \mathbf{N}$. If $(x_n)$ is any sequence in a set $X$, put $y_n = x_{f(n)}$. Show that the elementary filters associated with the sequences $(x_n)$ and $(y_n)$ are the same.

Deduce that if $(a_n)$ and $(b_n)$ are two sequences of elements of a set $X$ such that the filter associated with $(b_n)$ is finer than the filter associated with $(a_n)$, then the filter associated with $(b_n)$ is the same as the filter associated with some subsequence of $(a_n)$.