Real Numbers form Perfect Set

Theorem

Consider the set of real numbers $\R$ as a (complete) metric space with the usual (Euclidean) metric.

Then $\R$ forms a perfect set.

Proof

By definition, a perfect set is a set which equals its set of limit points.

Let $x \in \R$.

Consider the sequence:

$\sequence {y_k} = x + \dfrac 1 k$

Then as $\sequence {z_k} = \dfrac 1 k$ is a basic null sequence it follows that:

$\ds \lim_{n \mathop \to \infty} \sequence {y_k} = x$

Thus we see that $x$ is a limit point of $S$.

$\blacksquare$