# Thread: Prove that this converges

1. ## Prove that this converges

Let [a,b] be a closed interval in $\displaystyle \mathbb R$, and suppose that {$\displaystyle \ s_n$}, $\displaystyle \ n \geq 1$ is a sequence within [a,b]. Prove that the sequence {$\displaystyle s_n/n$}, $\displaystyle \ n \geq 1$, converges.

Anyone know how to do this problem? thanks in advance guys.

2. Originally Posted by Cato
Let [a,b] be a closed interval in $\displaystyle \mathbb R$, and suppose that {$\displaystyle \ s_n$}, $\displaystyle \ n \geq 1$ is a sequence within [a,b]. Prove that the sequence {$\displaystyle s_n/n$}, $\displaystyle \ n \geq 1$, converges.

Anyone know how to do this problem? thanks in advance guys.
note that the $\displaystyle s_n$'s are bounded.

3. We have
$\displaystyle \frac{a}{n}\leq \frac{s_n}{n}\leq \frac{b}{n}$
then use squeeze theorem

4. Why $\displaystyle a/n$ and $\displaystyle b/n$

5. Originally Posted by Cato
Why $\displaystyle a/n$ and $\displaystyle b/n$
it is within the interval [a,b]. so the smallest an element of the sequence can be is a, while the largest it can be is b

6. Thanks guys