# Prove that this converges

• Oct 8th 2008, 07:30 PM
Cato
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.
• Oct 8th 2008, 07:37 PM
Jhevon
Quote:

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.
• Oct 8th 2008, 07:39 PM
watchmath
We have
$\displaystyle \frac{a}{n}\leq \frac{s_n}{n}\leq \frac{b}{n}$
then use squeeze theorem
• Oct 8th 2008, 08:09 PM
Cato
Why $\displaystyle a/n$ and $\displaystyle b/n$
• Oct 8th 2008, 08:11 PM
Jhevon
Quote:

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
• Oct 8th 2008, 08:23 PM
Cato
Thanks guys