Question about bound sequence and the limit

Question

We were taught that the limit of the sequence exists if the sequence is monotone and bound.

I understand how to tell if a sequence is monotonic, but I don't understand what "being bound" means. How would you tell if a sequence is bounded?

Answer 1

This criteria (theorem) is known as Monotonic Sequence Theorem for Convergence and there are not general rules to determine monotonicity and bounding of the sequence since it depends case by case.

Note that bounded means that $\exists m,M\in\mathbb{R}$ such that $\forall n$

$$m\le a_n \le M$$

A well known example is the sequence

$$\left(1+\frac1n\right)^n$$

for which we can prove monoticity, notably that it is strictly increasing, and bounding since

$$2\le\left(1+\frac1n\right)^n\le3$$

thus the limit exist and we know that it is the number $e$.