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An infinite sequence of real numbers (in blue). This sequence is neither increasing, decreasing, convergent, nor Cauchy.It is, however, bounded. In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters.
A bounded operator: is not a bounded function in the sense of this page's definition (unless =), but has the weaker property of preserving boundedness; bounded sets are mapped to bounded sets (). This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for ...
Every uniformly convergent sequence of bounded functions is uniformly bounded. The family of functions f n ( x ) = sin n x {\displaystyle f_{n}(x)=\sin nx} defined for real x {\displaystyle x} with n {\displaystyle n} traveling through the integers , is uniformly bounded by 1.
Corollary — If a sequence of bounded operators () converges pointwise, that is, the limit of (()) exists for all , then these pointwise limits define a bounded linear operator . The above corollary does not claim that T n {\displaystyle T_{n}} converges to T {\displaystyle T} in operator norm, that is, uniformly on bounded sets.
Bounded poset, a partially ordered set that has both a greatest and a least element; Bounded set, a set that is finite in some sense Bounded function, a function or sequence whose possible values form a bounded set; Bounded set (topological vector space), a set in which every neighborhood of the zero vector can be inflated to include the set
Every infinite sequence of real numbers has an infinite monotone subsequence (This is a lemma used in the proof of the Bolzano–Weierstrass theorem). Every infinite bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence (This is the Bolzano–Weierstrass theorem ).
Since F is uniformly bounded, the set of points {f(x 1)} f∈F is bounded, and hence by the Bolzano–Weierstrass theorem, there is a sequence {f n 1} of distinct functions in F such that {f n 1 (x 1)} converges. Repeating the same argument for the sequence of points {f n 1 (x 2)} , there is a subsequence {f n 2} of {f n 1} such that {f n 2 (x ...
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.