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

## Homework Statement

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.

## Homework Equations

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

## 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?

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?

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?

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.

Yes it does! Check the squeeze theorem carefully!

HallsofIvy
Homework Helper

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
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?

, but it doesnt say anything about if the limit of the sequence's terms is taken to infinity.