
convergent sequence
$\displaystyle prove \ that \ if \ a_n \ is \ an \ increasing \ sequence \ , \ and \ b_n \ is \ a \ decreasing$ $\displaystyle \ sequence \ with \ a_n \ \leq \ b_n \ , \ \forall n\in N \ then \ both$
$\displaystyle a_n \ and \ b_n \ are \ convergent \ sequences$
$\displaystyle \ \ with \ lim \ a_n \leq lim \ b_n$

Do you know that any bounded monotone sequence converges?
Do you understand how $\displaystyle b_1$ is an upper bound for the $\displaystyle (a_n)$ sequence?
What is a lower bound for the $\displaystyle (b_n)$ sequence?
Now put this together.

Suppose $\displaystyle a_n\to a,b_n\to b$ and $\displaystyle a>b$. Hence, $\displaystyle \exists N_1:aa_n>a\frac{a+b}{2},\forall n>N_1$ and $\displaystyle \exists N_2:b_nb>\frac{a+b}{2}b,\forall n>N_2$. So for any $\displaystyle n\geq \max\left\{N_1,N_2\right\}$ you have $\displaystyle a_n>b_n$.