Sequential Compactness

Real Analysis · Axiom Academy

A set is sequentially compact when every sequence inside it has a subsequence that converges to a point still inside — the most usable face of compactness. No matter what sequence (x_n) you drop into the set, you can always pull out a subsequence (x_ n_k ) that homes in on some limit — and crucially, that limit must live inside the set. … has a subsequence with a limit back in K 2. Success: the Closed Interval [0,1] The closed interval [0,1] is sequentially compact. By the Bolzano–Weierstrass theorem , any sequence trapped in a closed bounded interval must have a convergent subsequence. Take — it never settles as a whole. But pull out the "low" terms and that subsequence marches straight down to 0 , and . 3. Failure 1: the Open Interval (0,1) The open interval (0,1) is not sequentially compact. The trouble is that its only "destination" sits just outside the set. Take , that is . Every subsequence of it still converges to 0 — but . There is no subsequence whose limit lands back inside the set. 4. Failure 2: the Unbounded Ray Closedness alone is not enough. The ray is closed, yet it is still not sequentially compact — this time the sequence simply runs off to infinity. Take x_n = n , that is . Any subsequence of it is still unbounded and races to . A convergent sequence is bounded, so no subsequence can converge at all. 5. Why It All Works: the Equivalence For metric spaces, the abstract open-cover definition and the concrete sequence definition describe the same property .

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