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Course Material 2.5 Monotone Sequences

Definition

A real sequence {𝑥𝑛 } is said to be increasing if for each 𝑛 we have 𝑥𝑛 ≤ 𝑥𝑛+1 and {𝑥𝑛 } is
said to be decreasing if for each 𝑛 we have 𝑥𝑛 ≥ 𝑥𝑛+1 . The sequence {𝑥𝑛 } is said to be
monotone if it is either increasing or decreasing.

A sequence {𝑥𝑛 } is said to be strictly increasing if for each 𝑛 we have 𝑥𝑛 < 𝑥𝑛+1 and {𝑥𝑛 }
is said to be strictly decreasing if for each 𝑛 we have 𝑥𝑛 > 𝑥𝑛+1

Examples

1
1. {𝑛} is a decreasing sequence
1
2. {1 − 𝑛} is an increasing sequence
(−1)𝑛
3. { } is not a monotone sequence
𝑛


Note:

1. If sequence {𝑥𝑛 } is increasing then we have 𝑥1 ≤ 𝑥2 ≤ ⋯ . . 𝑥𝑛 ≤ 𝑥𝑛+1 … Hence
every increasing sequence is bounded below by its first term and similarly and
every decreasing sequence is bounded above by its first term.
2. An increasing sequence is bounded if and only if it is bounded above and a
decreasing sequence is bounded if and only if it is bounded below.

Remark: We know that if a sequence is convergent then it is bounded and the
converse need not be true. But in the case of monotone sequences, the converse is
also true.

Theorem.

A monotone sequence is convergent if and only if it is bounded.

Proof.

Since convergent sequences are bounded, in this case, it is enough to prove only that
a monotone sequence is convergent if it is bounded

Also, it is enough to prove the theorem for increasing sequences as the proof for
decreasing sequences is similar.