# If {S_n} is a sequence whose values lie inside [a,b], prove {S_n/n} is convergent.

1. Feb 1, 2012

### Shayes

1. The problem statement, all variables and given/known data

If {S_n} is a sequence whose values lie inside an interval [a,b], prove {S_n/n} is convergent.

We don't know Cauchy sequence yet. All we know is the definition of a bounded sequence, and convergence and divergence of sequences. Along with comparison tests and Squeeze theorem.

2. Relevant equations

Limit of a sequence: abs(S_n - L) < Epsilon whenever n>=N, provided for Epsilon>0.

3. The attempt at a solution

I see that every convergent sequence is bounded, but the opposite isn't always true, so how do I show that the smaller sequence is convergent given that {S_n} is bounded?

2. Feb 1, 2012

### susskind_leon

Re: If {S_n} is a sequence whose values lie inside [a,b], prove {S_n/n} is convergent

I think you can use the squeeze theorem, since a/n <= S_n/n <= b/n and they both converge to 0. Do you need to proove that a/n converges or can you use that?

3. Feb 1, 2012

### Shayes

Re: If {S_n} is a sequence whose values lie inside [a,b], prove {S_n/n} is convergent

i thought of that, but is that the same as saying the entire sequence converges?

i feel like this statement says that each term is squeezed between the interval, but it doesnt say anything about if the limit of the sequence's terms is taken to infinity.

4. Feb 2, 2012

### susskind_leon

Re: If {S_n} is a sequence whose values lie inside [a,b], prove {S_n/n} is convergent

Yes it does! Check the squeeze theorem carefully!

5. Feb 2, 2012

### HallsofIvy

Staff Emeritus
Re: If {S_n} is a sequence whose values lie inside [a,b], prove {S_n/n} is convergent

I have no idea what this means, but the crucial point is the "n" in the denominator. What are the linits of a/n and b/n?