Section 8*: Problem 1 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. The sum $\sum_{k=1}^{n}b_{k}$ is defined by induction as follows: $\begin{aligned} & \sum_{k=1}^{n}b_{k}=b_{1} & & \mbox{for }n=1,\\ & \sum_{k=1}^{n}b_{k}=(\sum_{k=1}^{n-1}b_{k})+b_{n} & & \mbox{for }n>1. \end{aligned}$ Let $A$ be the set of real numbers; choose $\rho$ so that Theorem 8.4 applies to define this sum rigorously. We sometimes denote the sum $\sum_{k=1}^{n}b_{k}$ by the symbol $b_{1}+b_{2}+\cdots+b_{n}$ .

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

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Section 8*: Problem 1 Solution

Section 8*: Problem 1 Solution