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:
- $\sequence {y_k} = x + \dfrac 1 k$
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 $S$.
$\blacksquare$