# Proving a sequence converges

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1. Oct 13, 2015

### Matt B.

1. The problem statement, all variables and given/known data: Prove that if xn is a sequence such that |xn - xn+1| ≤ (1/3n), for all n = 1,2,..., then it converges.

2. Relevant equations: The definition of convergence.

3. The attempt at a solution: I attempted to prove this by induction, so I am clearly far off the mark here. Any advice would be appreciated.

2. Oct 13, 2015

### 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.

3. Oct 13, 2015

### Matt B.

Our definition of convergence provided is the following:

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

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

4. Oct 13, 2015

### Ray Vickson

Have you ever heard of Cauchy sequences?

5. Oct 13, 2015

### Matt B.

Yes.

6. Oct 13, 2015

### Ray Vickson

OK.... so?

7. Oct 13, 2015

### Matt B.

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

8. Oct 13, 2015

### Ray Vickson

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.