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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.
It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges. Let S be a nonempty set of real numbers. If S has exactly one element, then its only element is a least upper bound. So consider S with more than one element, and suppose that S has an upper bound B 1.
Proof: (sequential compactness implies closed and bounded) Suppose A {\displaystyle A} is a subset of R n {\displaystyle \mathbb {R} ^{n}} with the property that every sequence in A {\displaystyle A} has a subsequence converging to an element of A {\displaystyle A} .
These last two properties, together with the Bolzano–Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and the Heine–Borel theorem. Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence ...
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.
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:
So let f be such arbitrary bounded continuous function. Now consider the function of a single variable g(x) := f(x, c). This will obviously be also bounded and continuous, and therefore by the portmanteau lemma for sequence {X n} converging in distribution to X, we will have that E[g(X n)] → E[g(X)].
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.