# Weakly monotonic sequences

I am confused with the definition of 'Weakly Monotonic Sequences'. Which one is correct?

1. A sequence which is neither increasing nor decreasing
2. A sequence which is not increasing or not decreasing

So, if the 2nd one is the correct one, then does it imply that an increasing sequence is weakly monotonic?(Similarly, does it imply that a decreasing sequence is weakly monotonic?)

Can anyone please help me to understand the correct definition?

• I read your second definition differently than I think you meant it to be read. An increasing sequence has $a_{n+1}>a_n$ for all $n$. Now consider the sequence $\{(-1)^n\},$ for example: $(-1)^3<(-1)^2,$ so this sequence fails to satisfy the definition of "increasing sequence," and I would say it is "a sequence that is not increasing." Likewise I would say it is not decreasing. But it is not weakly monotonic. In other words, I think you wanted "not increasing" to mean $\forall n.\neg(a_{n+1}>a_n),$ but I think it means $\neg\forall n.(a_{n+1}>a_n),$ which is quite different. – David K Dec 14 '14 at 17:52

A sequence which is either weakly increasing or weakly decreasing, Weakly, i.e. $a_n\leq a_{n+1}$ (or $a_n\geq a_{n+1}$).

• So, is it same as monotonically increasing/decreasing? – user184652 Oct 28 '14 at 0:35
• @user184652 ...or monotonically decreasing. – Przemysław Scherwentke Oct 28 '14 at 0:37

Definition: A sequence (or function) $a(n)$ is increasing (decreasing) if $n\leq m$ implies $a(n)\leq a(m)$ ($n\leq m$ implies $a(n)\geq a(m)$. A monotone sequence (or function) is one that is either increasing or decreasing. With the exception of constant sequences (functions), these are mutually exclusive options.

Issues: It is common to give the above definition with strict inequalities ($<$ or $>$) instead of weak inequalities ($\leq$ or $\geq$). The difference usually does not matter, but it is a real difference. Because both definitions are common, a popular way of avoiding ambiguity is to use the terms strict and weak to specify whether you mean the definition with strict inequalities or the one with weak inequalities.

Examples:

$1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}\ldots$ is weakly decreasing, strictly decreasing, weakly monotone, and strictly monotone. It is not increasing.

$1,1,1,1,1,\ldots$ is weakly increasing, weakly decreasing, and weakly monotone. It is not strictly monotone (thus not strictly increasing or strictly decreasing).

In the abstract, if I were talking about decreasing sequences then both of these would fall under the discussion. If I want to exclude the second sequence as an option, then I will talk about strictly decreasing sequences.

Many other mathematicians would only be referring to the former behavior when talking about decreasing sequences. They would specifically say they are talking about weakly decreasing sequences if they wanted the latter sequence to also fall under the discussion.

Both of these uses of the term are common in textbooks and, in my experience you won't have trouble finding mathematicians in the same department who disagree on which is the "right" definition.