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Definition: A set is sequentially compact if every sequence {} in has a convergent subsequence converging to an element of . Theorem: A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}} is sequentially compact if and only if A {\displaystyle A} is closed and bounded.
Every bounded-above monotonically nondecreasing sequence of real numbers is convergent in the real numbers because the supremum exists and is a real number. The proposition does not apply to rational numbers because the supremum of a sequence of rational numbers may be irrational.
However, bounded and weakly closed sets are weakly compact so as a consequence every convex bounded closed set is weakly compact. As a consequence of the principle of uniform boundedness, every weakly convergent sequence is bounded. The norm is (sequentially) weakly lower-semicontinuous: if converges weakly to x, then
In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in .. Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice).
Every bounded sequence in has a convergent subsequence, by the Bolzano–Weierstrass theorem. If these subsequences all have the same limit, then the original sequence also converges to that limit. If these subsequences all have the same limit, then the original sequence also converges to that limit.
For a general class of spaces, similarly to weak convergence, every bounded sequence has a Delta-convergent subsequence. Delta convergence was first introduced by Teck-Cheong Lim, [ 1 ] and, soon after, under the name of almost convergence, by Tadeusz Kuczumow.
A sequence is convergent if and only if every subsequence is convergent. If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point. These properties are extensively used to prove limits, without the need to directly use the cumbersome formal definition.
The monotone convergence theorem (described as the fundamental axiom of analysis by Körner [1]) states that every nondecreasing, bounded sequence of real numbers converges. This can be viewed as a special case of the least upper bound property, but it can also be used fairly directly to prove the Cauchy completeness of the real numbers.