Does the function a(n) = 0 increase and decrease; n is an element of the naturals?

[The Student]

Does the function a(n) = 0 increase and decrease; n is an element of the naturals?

I ask this because a(n) = 0(1), 0(1), 0(2) ... a(n), a(n+1) fits the definition of an increasing limit where a(n) < or = to a(n+1), and it fits the definition of a decreasing sequence where a(n) > or = a(n+1). If the answer is "yes", then do we call it something else even though it increases and decreases?

 

[daon2]

The sequence does not increase nor decrease. A constant sequence is just that, constant. Sometimes sequences which don't decrease are referred to as non-decreasing (as opposed to increasing). Constant sequences are both non-increasing and non-decreasing.

 

[The Student]

The sequence does not increase nor decrease. A constant sequence is just that, constant. Sometimes sequences which don't decrease are referred to as non-decreasing (as opposed to increasing). Constant sequences are both non-increasing and non-decreasing.

It's frustrating that the constant follows the definition of an increasing or decreasing monotone sequence but then is not considered to be either.

 

[daon2]

It's frustrating that the constant follows the definition of an increasing or decreasing monotone sequence but then is not considered to be either.

An increasing sequence is one that increases: \(\displaystyle a_n < a_{n+1}\) for all \(\displaystyle n\). I don't doubt that some authors might abuse or differently interpret the word "increase" but the textbook I use and those that I learned from have it as a strict inequality.

I have seen "monotonically increasing" used in the weaker sense you describe. But that is not the same as "increasing."

 

[HallsofIvy]

IF your text book gives, as the definition of "increasing", "a(n) < or = to a(n+1)" then, yes, that, and any constant sequence, is an "increasing" sequence. Further, if your text gives, as the definition of "decreasing", "a(n)> or = a(n+ 1)", then any constant sequence is also a "decreasing sequence".

Further, the only sequences that are both "increasing" and "decreasing" are the constant sequence.

You should however be aware that this NOT a standard terminology. Many texts define a sequence to be "increasing" only if "a(n)< a(n+1)", and "decreasing" only if "a(n)> a(n+1)". With that convention, constant sequences are neither increasing nor decreasing.

Text books that use "increasing" to mean "a(n)< or = a(n+1)" and "decreasing" to mean "a(n)> or = a(n+1)" typically use the phrase "strictly increasing" for sequences such that "a(n)< a(n+1)" and "strictly decreasing" for sequences such that "a(n)> a(n+1)"