Waste less time on Facebook — follow Brilliant.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Topology
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Is the sequence \(\{a_n\}_{n=1}^{\infty}\) given by \(a_n=\frac{1}{n}\) a Cauchy sequence?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Is the sequence \(\{a_n\}_{n=1}^{\infty}\) given by \[a_n=\left\{ \begin{aligned} &\frac{1}{n}&&\text{ if }n\text{ is even}\\ &1+\frac{1}{n}&&\text{ if }n\text{ is odd}\end{aligned}\right.\] a Cauchy sequence?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
How many of the following statements are true?
I. If \(a_n\) is a Cauchy sequence of rational numbers, then the limit of the \(a_n\) is a rational number.
II. If \(a_n\) is a Cauchy sequence of irrational numbers, then the limit of the \(a_n\) is an irrational number.
III. If \(a_n\) is a Cauchy sequence of real numbers, then the limit of the \(a_n\) is a real number.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If the sequence \(\{a_n\}_{n=1}^{\infty}\) is a Cauchy sequence, which of the following must also be Cauchy sequences? \[\text{I.} \left\{\frac{1}{a_n}\right\}_{n=1}^{\infty}\quad \text{II.} \left\{a_n^2\right\}_{n=1}^{\infty}\quad\text{III.} \left\{\sin a_n\right\}_{n=1}^{\infty}\]
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
How many of the following statements are true for a sequence of real numbers \(a_n\)?
I. If \[\lim_{n\to\infty} |a_n-a_m|=0\] for a single value of \(m\), then \(\{a_n\}_{n=1}^{\infty}\) is a Cauchy sequence.
II. If \[\lim_{n\to\infty} |a_n-a_m|=0,\] for every \(m\), then \(\{a_n\}_{n=1}^{\infty}\) is a Cauchy sequence.