- #1
sayan2009
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Homework Statement
given a set A(subset of R(reals)) is bounded.and for all x belongs to R there exists epsilon(eps) such that {(x-eps,x+eps) intersection A} is countable..to prove A is countable
A countable bounded set is a collection of elements that can be counted and has a finite upper bound or limit. This means that there is a maximum value or size for the elements in the set.
A countable bounded set has a finite upper bound, while an unbounded set does not have a limit or maximum value for its elements. This means that the elements in an unbounded set can continue indefinitely without reaching a maximum value.
An example of a countable bounded set is a set of natural numbers between 1 and 10. This set has a finite upper bound (10) and can be counted (1, 2, 3, 4, 5, 6, 7, 8, 9, 10).
No, a countable bounded set cannot be infinite. As mentioned, a countable bounded set has a finite upper bound, which means that the number of elements in the set is limited and cannot continue indefinitely.
The cardinality of a set refers to the number of elements in the set. A countable bounded set has a finite cardinality because the number of elements in the set is limited. This is in contrast to an unbounded set, which has an infinite cardinality.