Definition 1

Given a number $\varepsilon > 0$, a real number $r$ is said to be an approximation to within $\varepsilon$ to a real number $x$, if $|x - r| \leq \varepsilon$; $r$ is said to be an approximation by defect if $r \leq x$, by excess if $r \geq x$.

Let $A$ be a dense subset of $\mathbf{R}$. For each $x \in \mathbf{R}$ and each $\varepsilon > 0$ there is an approximation by defect (resp. excess) to $x$ to within $\varepsilon$ belonging to $A$, since the interval $]x - \varepsilon, x[$ (resp. $]x, x + \varepsilon[$) contains at least one point of $A$. If we now consider a given strictly decreasing sequence $(\varepsilon_n)$ of numbers $> 0$, tending to 0, and if $r_n$ is an approximation to $x$ to within $\varepsilon_n$, then the sequence $(r_n)$ has $x$ as limit as $n$ tends to infinity.

In the case where $A$ is a subgroup of the additive group $\mathbf{R}$, and we restrict the $\varepsilon_n$ to belong to $A$, we can define canonically for each $x \in \mathbf{R}$ a sequence $(r_n)$ of approximations to $x$ by defect, belonging to $A$.

For by Archimedes' axiom (§ 2, no. 1, Theorem 1), the set of integers $p$ such that $p \varepsilon_n \leq x$ has a greatest element $p_n$; in other words, there is a unique integer $p_n$ such that

Since $|x - p_n \varepsilon_n| \leq \varepsilon_n$, it follows that $p_n \varepsilon_n$ is an approximation to $x$ to within $\varepsilon_n$ by defect, and belongs to $A$ by hypothesis; similarly $(p_n + 1) \varepsilon_n$ is an approximation to $x$ to within $\varepsilon_n$ by excess, belonging to $A$, and the two sequences $(p_n \varepsilon_n)$ and $((p_n + 1) \varepsilon_n)$ have $x$ as their limit.