I recently begun to read Walter Rudin magnum upos "Principles of Mathematical Analysis" and i'm having a little trouble in understanding the proof of the the following statement:
2.41 Theorem: if a set $E$ in $R^k$ has one of the following three properties then it has the other two:
(a) $E$ is closed and bounded;
(b) $E$ is compact;
(c) Every infinite subset of $E$ has a limit point in $ E$.
He proves that (a) implies (b) and that (b) implies (c), remaining only to show that (c) implies (a). He started as follow:
If$ E$ is not bounded, then $E$ contains points $x_n$ with
$ |x_n| > n ( n = 1,2,3...) $
The set $S$ consisting of these points $x_n $ in infinite and clearly has no limit points in $R^k...$
My question is why this set $ S $ has no limit points in $R^k $?
Thanks in advance.