By Bolzano-Weierstrass Theorm we know that every bounded sequence has a limit point. I wanted to understand how a monotonic bounded sequence has a unique limit point?
Thanks a lot!
PS: I am familiar with the following proofs
- Montonic bounded sequences are convergent
- Convergent sequence has a unique limit point.
Using these two I can say that Monotonic bounded sequences have a unique limit point. However, I am looking for a simpler and more direct process through which I could directly show the required result
If $(x_n)_{n\in \mathbb{N}}$ is a non-decreasing monotonic sequence then you can write its limit as the supremum of its elements: $$ \lim _{n\to \infty} x_n = \sup _{n\in \mathbb{N}} x_n$$
It's a well known fact that the supremum of a subset of $\mathbb{R}$ is unique.
If the sequence is non-increasing monotonic then you just have to take the infinum.
(note that the result can be generalized in other spaces, for example for sequences with values in a totally ordered set)