Monotone Sequence

analysis

Definition

Monotonous Sequence

A sequence $(a_n{)}_{n{\ge }_k}$ is a repeatedly increase or repeatedly decreasing sequence.

Monotonously Increasing:

$$\forall i,j\ge k:i\le j⟹a_i\le a_j$$

Monotonously Decreasing:

$$\forall i,j\ge k:i\le j⟹a_i\ge a_j$$

Convergence

A monotonous sequence is convergent if and only if it is limited.

Let $(a_n{)}_{n\ge 0}$ a monotonously increasing sequence. From the convergence, it follows that the sequence is bounded. We only need to show that boundedness is sufficient for convergence.

According to the completeness theorem of the real numbers, $(a_n{)}_{\ge 0}$ has a supremum. Let $a=\sup a_n$ and let $\epsilon >0$. Since $a-\epsilon$ is not an upper bound of $(a_n{)}_{n{\ge }_0}$, there exists an $N(\epsilon )$ such that $a_{N(\epsilon )}>a-\epsilon$. Due to monotonicity, $a_n>a-\epsilon$ must also hold for all $n>N(\epsilon )$. Therefore, almost all $a_n$ lie in $U_{\epsilon }(a)$. Since $\epsilon$ can be chosen arbitrarily, it follows that ${\lim }_{n\to \infty }{a}_n=a$. $\square$