Exercise · ZW-CAUCHY-SEQUENCE-005 · ★☆☆ Basic
Which statement is true in \(\mathbb R\)?
Choose one option.
Show answer
- B. Every convergent sequence is Cauchy
Explanation
If \(a_n\to L\), for large \(m,n\) both terms are within \(\varepsilon/2\) of \(L\), hence within \(\varepsilon\) of each other. The others fail: \((-1)^n\) is bounded but not Cauchy, and \(\sqrt n\) has \(a_{n+1}-a_n\to0\) but is unbounded.
Review the concept: Cauchy sequence
A sequence \((a_n)\) is Cauchy if, for every \(\varepsilon>0\), there is an index \(N\) beyond which any two terms are less than \(\varepsilon\) apart.
\[\forall\varepsilon>0\ \exists N:\ m,n\ge N\Rightarrow|a_m-a_n|<\varepsilon\]
More practice problems: cauchy sequence
- ZW-CAUCHY-SEQUENCE-001 Which of these sequences is Cauchy?
- ZW-CAUCHY-SEQUENCE-002 For a_n=1/n, what is the smallest N such that |a_m-a_n|<0.01 whenever m,n≥ N?
- ZW-CAUCHY-SEQUENCE-003 If a_n=1/n, compute |a_10-a_20|.
- ZW-CAUCHY-SEQUENCE-004 Let a_n=1+((-1)^n)/(n). Compute |a_3-a_4|.
- ZW-CAUCHY-SEQUENCE-006 Let H_n=1+12+cdots+1n. What is the smallest value of |H_2n-H_n| over n≥1?
- ZW-CAUCHY-SEQUENCE-007 If |a_k+1-a_k|≤ 1/2^k for all k, then |a_m-a_n|<1/2^n-1 for m>n. What is the smallest N this bound…
- ZW-CAUCHY-SEQUENCE-008 Why is a_n=(-1)^n not Cauchy?
- ZW-CAUCHY-SEQUENCE-009 A Cauchy sequence (a_n) has a subsequence converging to L. What follows?