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

$\displaystyle a_n \ and \ b_n \ are \ convergent \ sequences$

$\displaystyle \ \ with \ lim \ a_n \leq lim \ b_n$