Exercise 5, § 6

General Topology, III, 316

A rectangle is a subset of $\mathbf{N} \times \mathbf{N}$ which is the product of two intervals of N. Given a finite subset E of $\mathbf{N} \times \mathbf{N}$ , let $\varphi(E)$ be the smallest number of mutually disjoint rectangles whose union is E. Let $(E_n)$ be an increasing sequence of finite subsets of $\mathbf{N} \times \mathbf{N}$ which cover $\mathbf{N} \times \mathbf{N}$ and are such that the sequence $(\varphi(E_n))$ is bounded. If $(x_n), (y_n)$ are two convergent series whose terms belong to a Hausdorff topological ring A, then we have

\lim_{n \to \infty} \sum_{(h, k) \in E_n} x_h y_k = \left( \sum_{n=0}^\infty x_n \right) \left( \sum_{n=0}^\infty y_n \right).

Construct an example of a sequence $(E_n)$ satisfying the conditions above and such that $E_{n+1} - E_n$ contains only one element for each n.

Solution

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