Section 8*: Problem 5 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

Show that there is a unique function $h:\mathbb{Z}_{+}\rightarrow\mathbb{R}_{+}$ satisfying the formula $\begin{aligned} & h(1)=3,\\ & h(i)=[h(i-1)+1]^{1/2} & & \mbox{for }i>1. \end{aligned}$

Let $a_{0}=3$ , and $\rho(f:\{1,...,n\}\rightarrow\mathbb{R}_{+})=\sqrt{f(n)+1}$ . $\rho$ is well-defined, as $f(n)+1>0$ . Then, according to Theorem 8.4, there is a unique $h$ satisfying the equations above.

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Section 8*: Problem 5 Solution

Section 8*: Problem 5 Solution