I was reading Theorem 3.6(a) - " If {$p_n$} is a sequence in a compact metric space X, then some sub-sequence of {$p_n$} converges to a point of X ", from Principles of Mathematical Analysis by W. Rudin [Third edition, page 51]. The proof considers the range of {$p_n$}, the set E. Now there are two cases possible :
Case 1: E is finite. This part of the proof is understood.
Case 2: E is infinite. Here I tried to use the following method of proof :
Here E is an infinite subset of the compact metric space X, and we know, "An infinite subset of a compact set has a limit point in the given compact set". Therefore E has a limit point p in metric space X (This part I followed from the book).
As p is a limit point of E, every neighborhood of p contains a point q $\epsilon$ E such that q $\neq$ p.
Now let us consider a real number $r_1$ > 0. Then there exists a neighborhood $N_{r_1}(p)$ of p of radius $r_1$, which contains some point $p_{n_1}$ in E,such that d(p,$p_{n_1}$) < $r_1$, as every neighborhood of p contains infinitely many points of E.
Similarly consider another real number $r_2$ such that $r_1$ > $r_2$ > 0. Then there exists a neighborhood $N_{r_2}(p)$ of p of radius $r_2$, which contains some point $p_{n_2}$ in E,such that d(p,$p_{n_2}$) < d(p,$p_{n_1}$).
In this way choosing $p_{n_1}$,$p_{n_2}$,$p_{n_3}$, ... we get a sub sequence {$p_{n_i}$} of {$p_n$}.
Now as for the choices of radii of neighborhoods r1 > r2 > r3 > ..... > 0, we can say d(p,pn1) > d(p,pn2) > d(p,pn3) > .... , and so for every $\epsilon$ > 0 there exists an integer N such that for every $n_i$ > N, d(p, $p_{n_i}$) < $\epsilon$. Therefore {$p_{n_i}$} converges to p.
P.S : Also I am not sure behind the intuition of the method used for proving this part in the book - If E is infinite, E has a limit point p $\epsilon$ X. Choose $n_1$ so that d(p,$p_{n_1}$) < 1. Having chosen $n_1$, ... , $n_{i-1}$, there is an integer $n_i$ $\gt$ $n_{i-1}$ such that d(p,$p_{n_i}$) < 1/i, Then {$p_{n_i}$} converges to p.
Is the intuition behind this method is to choose each $n_i$ such that d(p,$p_{n_i}$) $\lt$ 1. If this is the case, why a specific metric 1 is chosen instead of a generic real number ?
Please give me your insights.