Monotone Sequences of Real Numbers

Monotone Sequences of Real Numbers

We will now look at two new types of sequences, increasing sequences and decreasing sequences.

Definition: A sequence of real numbers $(a_n)$ is said to be increasing if $\forall n \in \mathbb{N}$ $a_n ≤ a_{n+1}$. Similarly, a sequence of real numbers $(a_n)$ is said to be decreasing if $\forall n \in \mathbb{N}$ $a_n ≥ a_{n+1}$. A sequence $(a_n)$ is said to be monotone or monotonic if it is either increasing or decreasing.

For example, consider the sequence $\left ( \frac{1}{n} \right ) = (1, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}, \frac{1}{n+1}, ... )$. We note that $\forall n \in \mathbb{N}$, $n < n+1$ and so $\frac{1}{n} > \frac{1}{n+1}$, and so this sequence is decreasing and hence monotone.

The following graph represents the first 10 terms of the monotonically decreasing sequence $\left ( \frac{1}{n} \right )$:

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One such example of an increasing sequence is the sequence $(n + 2)$. Clearly $\forall n \in \mathbb{N}$, $n + 2 < (n+1) + 2 = n + 3$ (since if not, then $n + 2 ≥ n + 3$ which implies that $0 ≥ 1$, which is a contradiction). The following graph represents the first 10 terms of the monotonically increasing sequence $(n + 2)$:

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From the definition of an increasing and decreasing sequence, we should note that EVERY successive term in the sequence should either be larger than the previous (increasing sequences) or smaller than the previous (decreasing sequences). Therefore the sequence $(1, 2, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...)$ cannot be considered a decreasing sequence as $1 = a_1 \not ≥ a_2 = 2$. From this, we will formulate the following definitions:

Definition: A sequence of real numbers $(a_n)$ is said to be ultimately increasing if for some $K \in \mathbb{N}$ we have that $\forall n ≥ K$ then $a_n ≤ a_{n+1}$. Similarly, a sequence of real numbers $(a_n)$ is said to be ultimately decreasing if for some $K \in \mathbb{N}$ we have that $\forall n ≥ K$ then $a_n ≥ a_{n+1}$. A sequence $(a_n)$ is said to be ultimately monotone or ultimately monotonic if for some $K \in \mathbb{N}$, if $n ≥ K$ then $(a_n)$ is either ultimately increasing or ultimately decreasing.

Consider the sequence $(n^2 - 4n + 3) = (0, -1, 0, 3, 8, ...)$. This is an ultimately increasing sequence, since for $n ≥ 2$ we have that $a_n ≤ a_{n+1}$. The following graph represents the first 7 terms of this ultimately increasing sequence:

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