I'm currently reading the Baby Rudin and I've trouble with understanding the proof of the theorem 2.41.
Theorem 2.41. If a set $E$ in $R^k$ has one of these three properties, 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 $E$.
In (c)->(a), the proof construct a set $S$. Assuming $E$ is not bounded, then $E$ contains points $x_n$ with $\left|x_n\right|>n(n=1,2,3,...)$. The proof then states "The set $S$ consisting of these points $X_n$ is infinite and clearly has no limit point in $R^k$, hence has none in $E$. "
My question is why $S$ has no limit point in $R^k$?