Theorem of bounded monotone sequences

The link between monotonicity, boundedness and the existence of the limit is expressed by a fundamental theorem, which guarantees convergence without having to compute the limit.

Theorem — Bounded monotone sequences

A monotone and bounded sequence always admits a finite limit. More precisely:

• $(a_n)$ increasing and bounded above $\Rightarrow$ $a_n\to\sup_n a_n$ (a finite number).
• $(a_n)$ decreasing and bounded below $\Rightarrow$ $a_n\to\inf_n a_n$.

Observation — A monotone sequence always has a limit

A monotone sequence always has a limit (finite or infinite): if it increases and is bounded above $\to$ finite limit; if it increases and is not bounded above $\to +\infty$. The behaviour is mirror-image for decreasing ones.

Links

Topics: Sequences
Concepts: Limit of a sequence · Monotonicity · Bounded sequence
Methods: Monotone sequence theorem
Skills: Prove