Abbott's Understanding Analysis question 6.2.14 There is already a question regarding exercise 6.2.14 from Stephen Abbott's Understanding Analysis in this site.  But it does not answer my question.  The Question is posed below, and my question below it.

My question is what exactly the notation $f_{1,k}$ means ?. From what I understand from the question, $f_{1,k}$ means a subsequence of $f_n$ evaluated at the point $x_1$ i.e. the first index indicated the data point at which the functions are evaluated and second index $k$ simply means that it is a subsequence  i.e. $k \subseteq [n]$. So in effect, $f_{1,k}$ is always evaluated at $x_1$. Then what exactly $f_{1,k}(x_2)$ means ?. 
 A: $(f_{n}(x_{1}))_{n}$ is a sequence of numbers. The author claims
that we can find a sequence $(n_{k})_{k}$ with $n_{1}\leq n_{2}\leq\cdots$ such that $(f_{n_{k}}(x_{1}))_{k}$ converges. 
Next, note that $(f_{n_{k}})_{k}$ is a sequence of functions. To simplify notation, the author defines $f_{1,k}=f_{n_{k}}$ (the $1$ is to stress that the sequence $(n_{k})_{k}$ was picked with respect to $x_{1}$). Therefore, the sequence of functions $(f_{n_{k}})_{k}$ can now be written $(f_{1,k})_{k}$.
Next, note that $(f_{1,k}(x_{2}))_{k}$ is a sequence of numbers corresponding to evaluating the functions $f_{1,k}$ at the point $x_{2}$.
A: By Bolzano-Weierstrass, every bounded sequence of real numbers contains a convergent subsequence.   Define $f_{m,k}$ to be that subsequence of $f_{m-1,k}(x_m)$ for each $m$, starting with $f_{2,m}:=$ a convergent subsequence of $f_{1,k}(x_2)$.
So,  the point is that $f_{1,k}$, though it is a convergent subsequence when evaluated at $x_1$, can also be evaluated at $x_2$ (or any $x$, for that matter).
I suggest reading (a) again carefully...  Abbott explains that the $(f_{n_k})$ are generated by the subsequence of real numbers gotten by evaluating the $(f_n)$ at $x_1$...  this is done by taking the index ($n_k$) of the function corresponding to the $n_k$th number in the subsequence...
