Sentence examples for a sequence of nonempty from inspiring English sources

where is a sequence of nonempty subsets.

Let be a sequence of nonempty closed convex subset of.

An indeterminate string (or, more simply, just a string) x="x[1..n] on an alphabet Σ is a sequence of nonempty subsets of Σ.

A sequence of nonempty, closed, convex sets converges to as that is, if and only if.

Let { C n } be a sequence of nonempty closed subset of E.

Let be a sequence of nonempty closed convex subset of a reflexive Banach space.

Proof Let { A n } be a decreasing sequence of nonempty subsets of C, with A n ∈ A ω ( C ).

Let { A n } n = 1 ∞ be a decreasing sequence of nonempty, closed, and bounded subsets of a Banach space X with lim n → ∞ μ ( A n ) = 0, where μ is the E-L measure of nonconvexity of X, and let A ∞ = ⋂ n = 1 ∞ A n. Then A ∞ = ⋂ n = 1 ∞ co ¯ A n. Definition 1.3.

By Lemma 1.2, A ∞ = ⋂ n = 1 ∞ co ¯ A n, where co ¯ A n n = 1 ∞ is a decreasing sequence of nonempty closed, and convex subsets of the weakly compact and convex set co ¯ C. The Šmulian theorem [6, Theorem V.6.2] then allows us to conclude that A∞ is nonempty.

Let { A n } n = 1 ∞ be a decreasing sequence of nonempty and closed subsets of co ¯ C with limn→∞μ(A n ) = 0.

For the proof that (iii) implies (iv), let { A n } n = 1 ∞ be a decreasing sequence of nonempty, closed, and bounded subsets of X such that limn→∞μ(A n ) = 0. Since A1 is bounded and { A n } n = 1 ∞ is decreasing, there exists λ > 0 such that B n = λ A n ⊂ B X n ∈ ℕ.