Give an example of i. A countable set of real numbers that is bounded above but not below. ii. An uncountable set of real numbers that is bounded and has no maximum element. iii. A bounded set of real numbers that has a maximum element but no minimum element. iv. A set of real numbers whose elements are all rational and whose supremum is irrational. v. A set of real numbers whose elements are all irrational and whose supremum is rational. 3. (a) Give an example of i. A sequence of real numbers that is not bounded. ii. A sequence of real numbers that is bounded and is not convergent. iii. A sequence of real numbers that is monotonic and is not convergent. iv. A sequence of real numbers that is convergent and is not monotonic. v. A sequence of real numbers that is strictly decreasing and is convergent.
Added by Raymond Y.
Step 1
A countable set of real numbers that is bounded above but not below: One example is the set of positive integers {1, 2, 3, ...}. This set is bounded above by any real number greater than or equal to 1, but it is not bounded below. ii. An uncountable set of real Show more…
Show all steps
Close
Your feedback will help us improve your experience
Lucas Finney and 56 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Give an example of a sequence of irrational numbers which converges to an integer limit.
Sri K.
Give an example of a sequence of rational numbers that converges to an irrational number
Ben B.
Give two examples of: (a)]A sequence $(x_{n})$ of irrational numbers having a limit $\lim x_n$ that is a rational number. [(b)] A sequence $(r_{n})$ of rational numbers having a limit $\lim r_{n}$ that is an irrational number.
Paul F.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD