Question

3. Let $a_n$ be a sequence. (a) Assume that for every $\varepsilon > 0$ there exists $N_{\varepsilon} > 0$ such that for every $n > N_{\varepsilon}$, $|a_{n+1} - a_n| < \varepsilon$. Is $a_n$ Cauchy? Prove it or provide a counterexample. (b) Assume that for every $n \in \mathbb{N}$, $|a_{n+1} - a_n| < 2^{-n}$. Is $a_n$ Cauchy? Prove it or provide a counterexample. What if $|a_{n+1} - a_n| < \frac{1}{n}$?

          3. Let $a_n$ be a sequence.
(a) Assume that for every $\varepsilon > 0$ there exists $N_{\varepsilon} > 0$ such that for every $n > N_{\varepsilon}$,
$|a_{n+1} - a_n| < \varepsilon$.
Is $a_n$ Cauchy? Prove it or provide a counterexample.
(b) Assume that for every $n \in \mathbb{N}$,
$|a_{n+1} - a_n| < 2^{-n}$.
Is $a_n$ Cauchy? Prove it or provide a counterexample. What if $|a_{n+1} - a_n| < \frac{1}{n}$?

3. Let an be a sequence.
(a) Assume that for every ε > 0 there exists Nε > 0 such that for every n > Nε,
|an+1 - an| < ε.
Is an Cauchy? Prove it or provide a counterexample.
(b) Assume that for every n ∈ℕ,
|an+1 - an| < 2^-n.
Is an Cauchy? Prove it or provide a counterexample. What if |an+1 - an| < (1)/(n)?

Added by Elisa N.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Nick Johnson

08/16/2021

Step 1

A sequence {an} is said to be Cauchy if for every & > 0, there exists N > 0 such that for all m, n > N, |am - an| < &. Show more…

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Please provide detailed steps so I can understand how to approach the problem (Real Analysis Class) 3. Let an be a sequence. (a) Assume that for every & > 0 there exists Ne > 0 such that for every n > Ns |an+1 - an|< e. Is an Cauchy? Prove it or provide a counterexample. (b) Assume that for every n e N, |an+1- an|<2-n Is an Cauchy? Prove it or provide a counterexample. What if |an+1 -- an|< ?