The set {2−k | k∈Z+} is bounded and countably infinite. ... An unbounded set of real numbers is necessarily infinite, but a bounded set can be of any size up to and including the cardinality of the entire set of real numbers.

Can infinite sets be bounded?

The set of all numbers between 0 and 1 is infinite and bounded. The fact that every member of that set is less than 1 and greater than 0 entails that it is bounded.

Is a countably infinite?

A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. ... Countably infinite is in contrast to uncountable, which describes a set that is so large, it cannot be counted even if we kept counting forever.