Sequences, Limits, and Topology, Non-Standardly

The last of the calculus-rebuilding pages of the NSA spine. We recast convergence, Cauchy-ness, Bolzano–Weierstrass, and compactness in the language of unlimited indices and halos, recovering the theorems of sequences-series.md and metric-spaces.md with proofs that replace $\epsilon$–$N$ juggling by a single application of the standard part (hyperreals.md §4).

References: Goldblatt, Lectures on the Hyperreals, ch. 6 & 10; Keisler, Elementary Calculus, ch. 5; Robinson, Non-standard Analysis, ch. 3–4.

1. Convergence via unlimited indices

A real sequence $a:\mathbb{N}\to \mathbb{R}$ extends by transfer to an internal hypersequence ${}^{\ast }a:{}^{\ast }\mathbb{N}\to {}^{\ast }\mathbb{R}$ (transfer-principle.md). Its behavior "at infinity" is read off from the unlimited terms $a_H$, $H\in {}^{\ast }\mathbb{N}\setminus \mathbb{N}$ (internal-sets.md §1).

Theorem. $\underset{n\to \infty }{\lim }{a}_n=L\in \mathbb{R}$ iff $a_H\approx L$ for every unlimited $H\in {}^{\ast }\mathbb{N}$ — equivalently $\mathrm{st}(a_H)=L$ for every unlimited $H$.

Proof. ($\Rightarrow$) Given real $\epsilon >0$, convergence provides $N$ with $|a_n-L|<\epsilon$ for all $n>N$; transfer to ${}^{\ast }\mathbb{N}$: $|a_H-L|<\epsilon$ for all $H>N$, in particular all unlimited $H$. As $\epsilon$ is arbitrary, $a_H\approx L$. ($\Leftarrow$) If convergence fails, some real $\epsilon >0$ has $|a_{n_k}-L|\ge \epsilon$ along a subsequence; the class $H=[n_k]$ is unlimited with $|a_H-L|\ge \epsilon$, contradicting $a_H\approx L$. $\square$

The algebra of limits (sequences-series.md) is now just the algebra of $\mathrm{st}$: $\mathrm{st}(a_H+b_H)=\mathrm{st}(a_H)+\mathrm{st}(b_H)$, etc. (hyperreals.md §4.1). Divergence to $+\infty$ is "$a_H$ is positive infinite for every unlimited $H$."

2. Cauchy sequences and completeness

Theorem. $(a_n)$ is Cauchy iff $a_H\approx a_K$ for all unlimited $H,K$.

Thus a hypersequence "bunches up" at infinity precisely when it is Cauchy. Over $\mathbb{R}$, every finite hyperreal has a shadow (hyperreals.md §4), so a Cauchy sequence's unlimited terms share a single standard part $L=\mathrm{st}(a_H)$, and §1 gives $a_n\to L$. This is the completeness of $\mathbb{R}$ (real-numbers.md) reduced to the standard part theorem — the Cauchy criterion is "all unlimited terms lie in one halo, whose center is the limit."

3. Bolzano–Weierstrass and limit points

Theorem (Bolzano–Weierstrass, nonstandard form). A real number $L$ is a subsequential limit of $(a_n)$ iff $a_H\approx L$ for some unlimited $H$. Consequently a bounded sequence has a convergent subsequence: boundedness makes every $a_H$ finite, so $L=\mathrm{st}(a_H)$ exists for any single unlimited $H$, and it is a subsequential limit.

The contrast with §1 is the crisp nonstandard picture of the whole subject:

So the set of subsequential limits is exactly the set of shadows of unlimited terms — one hyperreal snapshot replaces the $\mathrm{limsup}/\mathrm{liminf}$ apparatus of sequences-series.md.

4. Near-standard points and nonstandard compactness

A point $x\in {}^{\ast }{\mathbb{R}}^m$ is near-standard if $x\approx p$ for some standard $p$ — i.e. $x$ lies in the halo of a real point. The pivotal theorem (due to Robinson) recasts compactness entirely in terms of halos:

Theorem (nonstandard compactness). A set $K\subseteq {\mathbb{R}}^m$ is compact iff every point of ${}^{\ast }K$ is near-standard with shadow in $K$: $$\forall x\in {}^{\ast }K\exists p\in K:x\approx p.$$

Proof idea. Compact $=$ closed + bounded (metric-spaces.md, Heine–Borel). Bounded $\Rightarrow$ every $x\in {}^{\ast }K$ is finite, so $\mathrm{st}(x)$ exists; closed $\Rightarrow$ $\mathrm{st}(x)\in K$. Conversely, if $K$ is unbounded it has an infinite hyperreal point (not near-standard); if $K$ is not closed, a boundary point $p\notin K$ has hyperreal approximants $x\approx p$ in ${}^{\ast }K$ with shadow $p\notin K$. $\square$

This single equivalence powers slick proofs: EVT — a continuous $f$ on compact $K$ attains its sup because a maximizing grid node $x_M\in {}^{\ast }K$ is near-standard, and $\mathrm{st}(x_M)$ is the maximizer (infinitesimal-calculus.md §5); Heine–Cantor — uniform continuity is pointwise microcontinuity plus near-standardness of every ${}^{\ast }K$ point (§2 of infinitesimal-calculus.md).

The dictionary in one line. Topological finiteness ("compact") becomes "no point escapes to infinity or into a gap": every hyperreal point of the set is infinitely close to an honest point of the set. This is the nonstandard reading of the open-cover definition in topology-manifolds.md §1 and metric-spaces.md.

5. Series, briefly

A series ${\sum }_n{c}_n$ converges to $S$ iff its partial sums $s_n$ do, i.e. iff the hyperfinite partial sum $s_H={\sum }_{n=1}^H{}^{\ast }{c}_n\approx S$ for every unlimited $H$ (§1). The tail-smallness Cauchy criterion becomes ${\sum }_{n=H+1}^K{}^{\ast }{c}_n\approx 0$ for all unlimited $H<K$ (§2). This reuses the hyperfinite sums of internal-sets.md §4 and dovetails with the integral of nonstandard-integration.md — series and integrals are both shadows of hyperfinite sums.

Where this page is used

• The nonstandard compactness criterion underlies the existence proofs in infinitesimal-calculus.md and applications.md.
• Completes the parallel with the classical analysis spine (sequences-series.md, metric-spaces.md).

Next: Applications of Non-Standard Analysis — the method in action, and the physics payoff.