The Monotone Convergence Theorem

# The Monotone Convergence Theorem

Recall from the Monotone Sequences of Real Numbers that a sequence of real numbers $(a_n)$ is said to be monotone if it is either an increasing sequence or a decreasing sequence. Furthermore, a sequence $(a_n)$ is increasing if $\forall n \in \mathbb{N}$, $a_n ≤ a_{n+1}$, and a sequence is decreasing if $\forall n \in \mathbb{N}$, $a_n ≥ a_{n+1}$.

We will now look at an important theorem that says monotone sequences that are bounded will be convergent.

 Theorem 1 (The Monotone Convergence Theorem): If $(a_n)$ is a monotone sequence of real numbers, then $(a_n)$ is convergent if and only if $(a_n)$ is bounded.
• Proof: Let $(a_n)$ be a monotone sequence.
• $\Leftarrow$ There are two cases to consider. The first case is when $(a_n)$ is an increasing sequence, and the second case is when $(a_n)$ is a decreasing sequence.
• Case 1: Suppose that $(a_n)$ is an increasing sequence that is bounded. Since $(a_n)$ is bounded $\exists M \in \mathbb{R}$ such that $\forall n \in \mathbb{N}$ we have that $a_n ≤ M$. Now consider the set $\{ a_n : n \in \mathbb{N} \}$. Since the sequence $(a_n)$ is bounded, then this set is bounded. By The Completeness Property of the Real Numbers, this set has a supremum in $\mathbb{R}$, call it $L = \sup \{ a_n : n \in \mathbb{N} \}$.
• Let $\epsilon > 0$ be given. Since $L$ is the supremum of $\{ a_n : n \in \mathbb{N} \}$, then $L - \epsilon$ is not an upper bound to $\{ a_n : n \in \mathbb{N} \}$ and so $\exists a_N$ from this sequence such that $L - \epsilon < a_N$. Since $(a_n)$ is an increasing sequence, then for all $n ≥ N$ we have $a_N ≤ a_n$, and so:
(1)
\begin{align} L - \epsilon < a_N ≤ a_n ≤ L < L + \epsilon \end{align}
• Omitting the unnecessary parts of the inequality we see that for $n ≥ N$ we have $L - \epsilon < a_n < L + \epsilon$, and so $\mid a_n - L \mid < \epsilon$. Since $\epsilon > 0$ is arbitrary we have that $\lim_{n \to \infty} a_n = L$, that is $(a_n)$ is convergent to $L$.
• Case 2: Suppose that $(a_n)$ is a decreasing sequence that is bounded. Since $(a_n)$ is bounded $\exists m \in \mathbb{R}$ such that $\forall n \in \mathbb{N}$ we have that $m ≤ a_n$. Now consider the set $\{ a_n : n \in \mathbb{N} \}$. Since the sequence $(a_n)$ is bounded, then this set is bounded. This set must then have an infimum in $\mathbb{R}$, call it $L = \inf \{ a_n : n \in \mathbb{N} \}$.
• Let $\epsilon > 0$ be given. Since $L$ is the infimum of $\{ a_n : n \in \mathbb{N}$, then $L + \epsilon$ is not a lower bound to $\{ a_n : n \in \mathbb{N} \}$ and so $\exists a_N$ from this sequence such that $a_N < L + \epsilon$. Since $(a_n)$ is a decreasing sequence, then for all $n ≥ N$ we have $a_N ≥ a_n$, and so:
(2)
\begin{align} L -\epsilon < L ≤ a_n ≤ a_N < L + \epsilon \end{align}
• Omitting the unnecessary parts of the inequality we see that for $n ≥ N$ we have $L - \epsilon < a_n < L + \epsilon$ and so $\mid a_n - L \mid < \epsilon$. Since $\epsilon > 0$ is arbitrary we have that $\lim_{n \to \infty} a_n = L$, that is $(a_n)$ is convergent to $L$.
• In both cases, $(a_n)$ was convergent. $\blacksquare$

It is important to note that The Monotone Convergence Theorem holds if the sequence $(a_n)$ is ultimately monotone (i.e, ultimately increasing or ultimately decreasing) and bounded. Therefore, provided the sequence $(a_n)$ is bounded and that $\exists N \in \mathbb{N}$ such that $\forall n ≥ N$ then either $a_n ≤ a_{n+1}$ or $a_n ≥ a_{n+1}$, then $(a_n)$ is convergent. The proof of this fact can be obtained with a slight modification above.

 Corollary 1: If $(a_n)$ is an increasing and bounded sequence, then $\lim_{n \to \infty} a_n = \sup \{ a_n : n \in \mathbb{N} \}$ and if $(a_n)$ is a decreasing and bounded sequence, then $\lim_{n \to \infty} a_n = \inf \{ a_n : n \in \mathbb{N} \}$.

There is no proof to corollary 1 as it is proven in the Monotone Convergence Theorem above when we defined $L$ to be either the supremum (for increasing sequences) or the infimum (for decreasing sequences) in cases 1 and 2. We note that by The Uniqueness of Limits of a Sequence Theorem, a sequence that converges has a unique limit which we've already found.

What's nice is that The Monotone Convergence Theorem alongside Corollary 1 provide a nice way to determine whether a sequence converges based on the set of the sequence's terms.