# Real Numbers form Perfect Set

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## 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:

- $\left \langle{y_k}\right \rangle = x + \dfrac 1 k$

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

- $\displaystyle \lim_{n \to \infty} \left \langle{y_k}\right \rangle = x$

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

$\blacksquare$