Sequential compactness and total boundedness

A metric space is totally bounded if, for every positive epsilon, there is a finite list of centers in the space whose radius-epsilon balls cover it. The list can grow as epsilon shrinks; demanding one fixed finite list for all scales would be far stronger. For the interval [0,1], a sufficiently fine equally spaced grid gives such a cover. The strict ball inequality is easy to preserve by making the grid spacing smaller than epsilon. This property says that any fixed accuracy leaves only finitely many representative locations to consider.

Total boundedness implies boundedness for a nonempty space: use a finite cover at radius one, select a reference center, and take the largest of the finitely many center distances plus one. The reverse implication fails. An infinite set with the discrete metric has diameter one, yet a ball of radius one half contains only one point. No finite list of these balls covers the infinite set. Thus even a closed bounded complete metric space can fail total boundedness and compactness; finite-dimensional Euclidean intuition is using an additional geometric fact.

Given an infinite sequence in a totally bounded space, cover the space by finitely many radius-one-half balls. One ball contains infinitely many sequence indices. Retain those indices, then use a finite radius-one-quarter cover and retain an infinite subcollection in one of its balls. Repeat with radii tending geometrically to zero. Choose increasing indices diagonally through these nested infinite index collections. For each fixed selection stage, all sufficiently late chosen terms lie in its ball, so any two have distance at most twice its radius.

This produces a Cauchy subsequence. Completeness then supplies a limit in the space. Both ingredients are needed: total boundedness alone may send the subsequence toward a missing endpoint, as in (0,1). Conversely, sequential compactness forces total boundedness. If some radius admitted no finite cover, repeatedly choose a new point outside all previously chosen balls. The resulting points stay at least that radius apart and therefore cannot have a Cauchy, hence cannot have a convergent, subsequence.

In the sequence space l², let e_n have a one in coordinate n and zeros elsewhere. Every e_n has norm one, but the distance between any distinct pair is the square root of two. No subsequence is Cauchy. The closed unit ball of l² is closed, bounded, and complete, yet not sequentially compact. The escape occurs across infinitely many independent coordinate directions rather than by growing farther from the origin. This example is a reason functional analysis must distinguish bounded sets from compact sets carefully.

Sequential compactness itself implies completeness. A Cauchy sequence has a convergent subsequence by sequential compactness, and then the whole original sequence converges to the same limit: use a late subsequence term as a bridge between any late term and that limit. Thus the equivalence is not merely a convenient sufficient criterion. It pinpoints the two distinct ways compactness can fail: missing limits or an inability to cover all possible directions with finitely many small balls.