If $a_n$ is a non decreasing sequence and $b_n$ is a non increasing sequence, and $a_n \leq b_n$ for all $n$, this is this sufficient condition for concluding that they both converge to the same limit.
If no, what else is needed?
Convergence of non increasing and non decreasing sequences to the same limit
real-analysissequences-and-series
Best Answer
The given conditions only imply that $a_n$ and $b_n$ both converge. Their limits may not be equal. For example take $a_n=1-\frac 1 n$ and $b_n =2+\frac 1 n$. If you also know that $b_n-a_n \to 0$ you can conclude that the limts are equal.