# Monotone convergence theorem

 related topics {math, number, function} {rate, high, increase}

In mathematics, there are several theorems dubbed monotone convergence; here we present some major examples.

## Contents

### Theorem

If ak is a monotone sequence of real numbers (e.g., if ak ≤ ak+1,) then this sequence has a finite limit if and only if the sequence is bounded.[1]

### Proof

We prove that if an increasing sequence {an} is bounded above, then it is convergent and the limit is $\sup_n \{a_n\}$.

Since {an} is non-empty and by assumption, it is bounded above, therefore, by the Least upper bound property of real numbers, $c = \sup_n \{a_n\}$ exists and is finite. Now for every $\varepsilon > 0$, there exists aN such that $a_N > c - \varepsilon$, since otherwise $c - \varepsilon$ is an upper bound of {an}, which contradicts to c being $\sup_n \{a_n\}$. Then since {an} is increasing, $\forall n > N , |c - a_n| = c - a_n \leq c - a_N < \varepsilon$, hence by definition, the limit of {an} is $\sup_n \{a_n\}.$

### Remark

If a sequence of real numbers is decreasing and bounded below, then its infimum is the limit.