I am currently having some difficulty with problem 2.7.8 in Introduction to Topology by Theodore Gamelin and Robert Greene. The problem goes as follows
A family F of real-valued functions on a topological space X is equicontinuous if for each $\varepsilon$>0, there exists a neighborhood U of x such that |f(y) - f(x)|<$\varepsilon$ for all y $\in$ U and all f $\in$ F. Let {f$_k$}$_{k=1}^{\infty}$ be a bounded sequence of real-valued functions on a compact space X such that the {f$_k$} are equicontinuous. Show that there exists a uniformly convergent subsequence of {f$_k$}.
Hint: For each n $\geq$ 1, find points x$_{n1}$,..., x$_{nm_n}$ and open neighborhoods W$_{nj}$ of the x$_{nj}$ such that X = W$_{n1}$$\cup$...$\cup$W$_{nm_n}$ and |f$_k$(x$_{nj}$) - f$_k$(y)|<$\frac{1}{n}$ whenever y $\in$ W$_{nj}$, 1 $\leq$ j $\leq$ m$_n$, and k $\geq$ 1. Use a diagonalization procedure to find a subsequence {f$_{k_i}$} such that {f$_{k_i}$(x$_{nj}$)}$_{k=1}^\infty$ converges for each fixed n and j. Then show that {f$_{k_i}$} converges uniformly on X."
So I understand the first part in the hint, that is easy. What I am stuck on is using this "diagonalization procedure". I think I have seen this done once in a proof a while ago and that's it, so I am not really experienced on using this type of procedure. If I could get some guidance on how to do that specific part I would greatly appreciate it.
