Total boundedness and finite nets

Try

A bounded interval can be approximated by finitely many sample centers at any requested distance scale. Divide [0,1] into sufficiently short pieces and choose one point in each. Every point of the interval lies near a chosen center. Total boundedness abstracts this finite-net property. The number of centers may grow as the tolerance shrinks; no fixed finite set is required to work at every scale.

Predict

Predict whether ordinary boundedness guarantees such nets. In an infinite discrete metric space, every pair of distinct points is distance one apart. The whole space has diameter one, yet an open ball of radius 1/2 contains only one point. No finite collection covers the infinite space. Thus boundedness can coexist with infinitely many separated directions or points at a fixed resolution.

Experiment

Compare the scale dependence.

Space	Radius 2	Radius 1/2	Totally bounded?
Infinite discrete space	One ball covers	Infinitely many needed	No
[0,1] with ordinary distance	One ball covers	A finite net exists	Yes
R with ordinary distance	No finite cover	No finite cover	No

Explain why total boundedness quantifies over every positive radius. Success at one coarse scale is ordinary boundedness-like information, not the full property. For a finite interval net, centers must belong to the metric space being covered.

Understand

Total boundedness yields a Cauchy subsequence from any sequence. Cover the space by finitely many balls of radius 1/2; one contains infinitely many indexed terms. Retain those indices. Cover again by finitely many balls of radius 1/4 and retain infinitely many of the already retained indices in one ball. Repeat with shrinking radii. Choose increasing indices diagonally, one from each nested retained set.

For any fixed stage, all sufficiently late selected terms belong to that stage’s ball. Two such terms are less than twice its radius apart by the triangle inequality. Since the radii tend to zero, the selected subsequence is Cauchy. The chosen balls need not themselves be nested as sets; the retained index sets are nested, which is what the argument uses.

Apply

Conversely, if total boundedness fails, fix ε>0 for which no finite collection of open ε-balls covers. Choose x₁ arbitrarily, then choose x₂ outside its ball, x₃ outside the two previous balls, and so on. Every pair of selected points is at least ε apart. No subsequence can be Cauchy. This construction will help connect total boundedness with sequential compactness.

Repair “a subset inherits the same net centers automatically.” Ambient centers may lie outside the subset. Instead cover the ambient space with ε/2-balls and choose one subset point from each ball that meets the subset. Points sharing that original ball are less than ε apart, so the new centers give a finite ε-net inside the subset.

Continue

Optional depth contrasts finite approximation with finite representation. A totally bounded infinite space can be approximated at every scale, yet no finite net identifies every point exactly. The distinction matters for numerical discretization: a grid resolution may control input distance, but output error additionally needs continuity or a quantitative bound.

Save the Cauchy-subsequence proof with retained indices and the factor two explicit. Next, completeness will turn that extracted Cauchy subsequence into a limit inside the space, and an open-cover argument will show why the combination is exactly compactness for metric spaces.