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In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ Y ⊆ X. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(X) (i.e., sequences of subsets of X). There are two common ways to define the limit of sequences of sets. In both cases:
A set with an upper (respectively, lower) bound is said to be bounded from above or majorized [1] (respectively bounded from below or minorized) by that bound. The terms bounded above ( bounded below ) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.
In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers ... converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums ...
A real-valued function is bounded if and only if it is bounded from above and below. [ 1 ] [ additional citation(s) needed ] An important special case is a bounded sequence , where X {\displaystyle X} is taken to be the set N {\displaystyle \mathbb {N} } of natural numbers .
Hypogeal germination (from Ancient Greek ὑπόγειος [hupógeios] 'below ground', from ὑπό [hupó] 'below' and γῆ [gê] 'earth, ground') is a botanical term indicating that the germination of a plant takes place below the ground. An example of a plant with hypogeal germination is the pea (Pisum sativum). The opposite of hypogeal is ...
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
Epigeal vs. hypogeal germination. Epigeal germination (Ancient Greek ἐπίγαιος [epígaios] 'above ground', from ἐπί [epí] 'on' and γῆ [gê] 'earth, ground') is a botanical term indicating that the germination of a plant takes place above the ground. An example of a plant with epigeal germination is the common bean (Phaseolus ...
There is a symmetric statement for submartingales with bounded expectation of the positive part. A supermartingale is a stochastic analogue of a non-increasing sequence, and the condition of the theorem is analogous to the condition in the monotone convergence theorem that the sequence be bounded from below.