习题 15, § 6

令$n \to f(n)$为$\mathbf{N}$到$\mathbf{N}$的映射，使得$f^{-1}(m)$对于每个$m \in \mathbf{N}$都是有限的。如果$(x_n)$是集合$X$中的任意序列，则放入$y_n = x_{f(n)}$。证明与序列 $(x_n)$ 和 $(y_n)$ 关联的基本滤波器是相同的。

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 $(a_n)$.