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By the least-upper-bound property, S has a least upper bound c ∈ [a, b]. Hence, c is itself an element of some open set U α, and it follows for c < b that [a, c + δ] can be covered by finitely many U α for some sufficiently small δ > 0. This proves that c + δ ∈ S and c is not an upper bound for S. Consequently, c = b.
A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set. An example of a set that lacks the least-upper-bound property is Q , {\displaystyle \mathbb {Q} ,} the set of rational numbers.
The sum a + b is the least upper bound of a and b: we have a ≤ a + b and b ≤ a + b and if x is an element of A with a ≤ x and b ≤ x, then a + b ≤ x. Similarly, a 1 + ... + a n is the least upper bound of the elements a 1, ..., a n. Multiplication and addition are monotonic: if a ≤ b, then a + x ≤ b + x, ax ≤ bx, and; xa ≤ xb ...
In particular, every positive real number is the least upper bound of the truncations of its infinite decimal representation; for example, is the least upper bound of {,,,, …}. A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations , and, in particular, with multiplication.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
13934 and other numbers x such that x ≥ 13934 would be an upper bound for S. The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on ...
Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.
On the other hand, / is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal between / and , and if / < / then is not infinitesimal. But 1 / ( 4 n ) < c / 2 {\displaystyle 1/(4n)<c/2} , so c / 2 {\displaystyle c/2} is not infinitesimal, and this is a contradiction.