Section 8*: Problem 2 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $(b_{1},b_{2},\ldots)$ be an infinite sequence of real numbers. We define the product $\prod_{k=1}^{n}b_{k}$ by the equations $\begin{aligned} & \prod_{k=1}^{1}b_{k}=b_{1},\\ & \prod_{k=1}^{n}b_{k}=(\prod_{k=1}^{n-1}b_{k})\cdot b_{n} & & \mbox{for }n>1. \end{aligned}$ Use Theorem 8.4 to define this product rigorously. We sometimes denote the product $\prod_{k=1}^{n}b_{k}$ by the symbol $b_{1}b_{2}\cdots b_{n}$ .

Let $a_{0}=b_{1}$ and $\rho(f:\{1,\ldots,n\}\rightarrow\mathbb{R})=f(n)\cdot b_{n+1}$ , then $\prod_{k=1}^{1}b_{k}\overset{def}{=}h(1)=b_{1}$ , and $\prod_{k=1}^{n}b_{k}\overset{def}{=}h(n)=\rho(h|\{1,...,n-1\})=h(n-1)\cdot b_{n}\overset{def}{=}(\prod_{k=1}^{n-1}b_{k})\cdot b_{n}$ .

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Section 8*: Problem 2 Solution

Section 8*: Problem 2 Solution