Does This Sequence Converge Given the Conditions?

[Matt B.]

Homework Statement

: [/B]Prove that if x_n is a sequence such that |x_n - x_n+1| ≤ (1/3ⁿ), for all n = 1,2,..., then it converges.

Homework Equations

: [/B]The definition of convergence.

The Attempt at a Solution

:[/B] I attempted to prove this by induction, so I am clearly far off the mark here. Any advice would be appreciated.

 

[andrewkirk]

A good start would be to write down your definition of convergence. All the hints that are likely to be needed are in that definition.

 

[Matt B.]

Our definition of convergence provided is the following:

{a_n} converges to A if ∀ε>0, ∃a positive integer N such that n ≥ N => | a_n - A | < ε.

I'm not sure how I am supposed to use this in the example above?

 

[Ray Vickson]

Our definition of convergence provided is the following:

{a_n} converges to A if ∀ε>0, ∃a positive integer N such that n ≥ N => | a_n - A | < ε.

I'm not sure how I am supposed to use this in the example above?

Have you ever heard of Cauchy sequences?

 

[Matt B.]

Have you ever heard of Cauchy sequences?

Yes.

 

[Ray Vickson]

Yes.

OK... so?

 

[Matt B.]

OK... so?

Do I just assume that x_n can be substituted for the usual x_n and that x_n+1 be substituted for x_m and attempt to find the n,n+1 ≥ N such that | x_n - x_n+1 | < ε, given ε > 0?

 

[Ray Vickson]

Do I just assume that x_n can be substituted for the usual x_n and that x_n+1 be substituted for x_m and attempt to find the n,n+1 ≥ N such that | x_n - x_n+1 | < ε, given ε > 0?

No, that is not what a Cauchy sequence is all about. A sequence is Cauchy if, given any $\epsilon > 0$ there is an $N = N(\epsilon)$ such that $|x_n - x_m|< \epsilon$ for all $n,m > N$. Note that this is $|x_n - x_m|$, not just $|x_n - x_{n+1}|$.

You might guess that if I mention Cauchy sequences, that must be for a good reason.