= Discrete subsets of the real line are countable
{title2=$B\subseteq\mathbb R\text{ discrete}\Rightarrow|B|\le\aleph_0$}
Enumerate all intervals with rational endpoints, using the countability of the <rational numbers> and listing pairs of enumeration indices by increasing sum. If every point $b$ of $B$ is an <isolated point>, some such interval contains $b$ and no other point of $B$. Assign to $b$ the least index of such an interval. Distinct points cannot have the same index, so this gives an <injective function> into $\mathbb N$ and proves that $B$ is a <countable set>. The argument uses isolation in the <subspace topology>, not closedness: the <discrete subset> $\{1/n:n\ge1\}$ illustrates that an accumulation point outside $B$ is allowed.
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