My question pertains to BBFSK, Vol I, Pages 143 and 144.
The following appears in the context of developing the real numbers as limits of sequences of rational numbers.
It is also easy to prove directly that the Cauchy criterion for convergence is valid for a sequence of real real numbers $\lim a^{\left(n\right)}\left(n=1,2,\dots\right).$ For let us determine, corresponding to each value of $n$, a natural number $n^{\prime}$ such that $\left|a_{n^{\prime}}^{\left(n\right)}-\lim a^{\left(n\right)}\right|<1/n.$ Then, from the hypothesis of the Cauchy criterion that for $\epsilon>0$ there exists a natural number $n_{0}$ such that $\left|\lim a^{\left(n\right)}-\lim a^{\left(m\right)}\right|<\epsilon$ for $n,m\ge n_{0},$ we see that the sequence $b$ of $b_{n}=a_{n^{\prime}}^{\left(n\right)}$ is fundamental and that $\lim b$ is the limit of the sequence of the numbers $\lim a^{\left(n\right)}.$
I do not understand the meaning of $n^{\prime}.$ Is it a function $n^{\prime}=\eta\left[n\right]:\mathbb{N}\to\mathbb{N}$ which changes value in some unspecified way as $n$ increases; that is $b_{n}=a_{\eta\left[n\right]}^{\left(n\right)}?$ Assuming that to be the case, then are the $b_{n}=a_{n^{\prime}}^{\left(n\right)}$ necessarily rational numbers, or necessarily real numbers?
Unfortunately, there's really no other example of the use of this index notation in the book.
The only thing that comes to mind when I read that is: don't try this in Vagus, or you'll get kicked out of the casino.