Question about monotonic sequences.

[cragar]

Homework Statement

If I have a constant sequence a,a,a,a,a,a, Is that monotonic?
because I have conflicting definitions some say that it has to increase or decrease,
but some say it just has to not increase or not decrease.
And also does every bounded sequence have monotonic sub sequence.

The Attempt at a Solution

I think that every bounded sequence has a monotonic sub sequence. Because I would eventually have to have an infinite amount of points that were the same or an infinite amount that were increasing or decreasing.

 

[LCKurtz]

Homework Statement

If I have a constant sequence a,a,a,a,a,a, Is that monotonic?
because I have conflicting definitions some say that it has to increase or decrease,
but some say it just has to not increase or not decrease.
And also does every bounded sequence have monotonic sub sequence.

The Attempt at a Solution

I think that every bounded sequence has a monotonic sub sequence. Because I would eventually have to have an infinite amount of points that were the same or an infinite amount that were increasing or decreasing.

I think most texts allow equality in the definition of monotone sequence and call the sequence "strictly monotone" for the case where strict inequality is used in the definition.

About your last question, I think so too, but can you prove it? Do you know the Bolzano Weierstrass theorem? It might help you construct an argument.