Exercise 1.12

Proof. First, each $d_i(i=1,2)$ induces a topology ${\tau }_i$ whose open balls are all

Next, remark that the monotonically increasing mapping $\varphi :R\to ]-1,1[$ is odd and that

$\varphi$ is therefore a ${\tau }_1$-homeomorphsim of $R$ onto $]-1,1[$. A first consequence is that, at fixed $a\in R$, given any positive scalar $𝜀$, the ${\tau }_1$-continuousness of $\varphi$ supplies an open ball $B_1(a,\eta )$ on which $|\varphi (a)-\varphi |<𝜀$. In terms of balls $B_i$, this reads as follows,

The second consequence is that the ${\tau }_1$-continuousness of ${\varphi }^{-1}$ yields similar inclusions

provided ${\eta }^{\prime }>0$. At arbitrary $𝜀$, the special case ${\eta }^{\prime }=\eta$ is the concatenation

which proves that ${\tau }_1={\tau }_2$. Finally, all inequalities $n<i<j$ over $N$ together yield

The sequence $n=0,1,2,\dots$ is therefore ${\tau }_2$-Cauchy. We will nevertheless establish that it ${\tau }_2$-diverges. To do so, we start by offering the ${\tau }_2$-converge to some $\lambda$: The triangle inequality immediately dismiss that assumption, as follows,

We then conclude that $d_2$ fails to be complete. □