Reciprocals of the positive integers decrease to zero

Complete ordered field (least-upper-bound property): every nonempty subset of an ordered field that is bounded above has a least upper bound.

The natural numbers $\mathbb{N}$ (von Neumann), Addition of natural numbers, Addition is commutative, Left identity for addition, and Order on the natural numbers: $\mathbb{N}$ is the von Neumann set of naturals, its successor is $\sigma (n)=n+1$, and $m\le n$ means $m+k=n$ for some $k\in \mathbb{N}$. Since $1+n=n+1=\sigma (n)$, every successor is at least $1$. Natural indices remain in $\mathbb{N}$; in field expressions the successor is carried into $\mathbb{R}$ by the canonical natural map of The canonical natural $\iota (n)=n\cdot 1_F$ of a field.

The canonical natural $\iota (n)=n\cdot 1_F$ of a field and Canonical naturals are positive and strictly increasing: write ${\iota }_{\mathbb{N}}:\mathbb{N}\to \mathbb{R}$ for the canonical natural map, with ${\iota }_{\mathbb{N}}(0)=0$ and ${\iota }_{\mathbb{N}}(\sigma (n))={\iota }_{\mathbb{N}}(n)+1$. It is positive on positive naturals and strictly increases there. Thus the displayed denominator $n+1$ means ${\iota }_{\mathbb{N}}(\sigma (n))$, not a von Neumann set used as a real number.

The unique embedding of ℚ into an ordered field: the rational embedding into $\mathbb{R}$ preserves order, so every rational $q>0$ has positive real image $\widehat{q}$. The multiplicative identity is positive, Inverses of positives are positive, and reciprocation reverses order, and Sign rules for products and monotonicity of multiplication supply $1>0$, positivity of the inverse of a positive real, and reversal of order by reciprocation on positive reals.

Limits and Cauchy sequences of reals: a real sequence converges to $0$ when, for every rational $q>0$, its absolute value is eventually less than $\widehat{q}$.