A list of numbers might keep getting bigger and bigger.
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I'm want to precisely define increasing for sequences.
A sequence a sub n is increasing if, whenever m is bigger
than n, then the mth term is bigger than the nth term.
This is capturing the idea that the terms are
getting bigger as I go further out in the sequence.
Let's take a look at an example of an increasing sequence.
For example the sequence a sub n equals n squared is an increasing sequence.
I can write out the first few terms.
The a sub one term is one and that's 4, 9, 16, right?
These are just the perfect squares and the claim is that this
is an increasing sequence, that as I go farther out in the sequence.
The terms get bigger and to really convince you of that formally, what I
need to show is that whenever m is bigger than
n, then a sub m is bigger than a sub n.
And indeed that's true because a sub m is m squared and a sub n is n squared.
And as long as these m and n are positive this is true whenever this is true.
We can also consider a sequence which is decreasing.
A sequence
a sub n is decreasing if whenever m is bigger than
n then a sub m is less than a sub n.
This is capturing the idea that the terms are getting smaller.
It's telling me that larger indices correspond to smaller terms.
So, as I go further out in the sequence, the terms are getting smaller.
It's easy to build examples of decreasing sequences.
Well, if, a sub-n is an increasing sequence So an example
of this would be an example we just saw an example would be a sub n equals n squared.
Well I've got an increasing sequence then the sequence b sub n
defined by just negating the terms of a sub n is a decreasing.
Sequence. Alright.
If the terms of a sub n are getting larger, the
terms of b sub n are getting more negative, they're getting smaller.
Why even bother with all of this?
Well, basically, I just want to give definitions for
the kinds of qualitative features that we might
be interested in, might be interested in a
sequence that's increasing or a sequence that's decreasing.
We might also be interested in a sequence which maybe isn't
necessarily getting bigger but at least it isn't getting any smaller.
Is nondecreasing if whenever m is bigger than n, then this isn't the case.
And to say that isn't the case means instead
of less than it's greater than or equal to.
So a non decreasing sequence is not getting any smaller
in the sense that future terms are at least as large as previous terms.
Here's an example of a non-decreasing sequence.
Sequence might start 1, 1, 2, 2, 3, 3, 4,
4 and so on where it sort of repeats itself.
And this sequence isn't increasing because later
terms are not larger than earlier terms.
two is not greater than two. But the sequence is non decreasing, right?
At least the terms aren't getting any smaller.
Not very surprisingly we can also define non-increasing.
A sequence is non-increasing if whenever m is bigger than n
then a sub m isn't bigger than a sub n. And isn't bigger means less
than equal or to. So a non-increasing sequence is, well, not
necessarily decreasing but at least its not getting any larger.
So thus far we've talked about
increasing and decreasing, non-increasing and non-decreasing.
Generally, it's just interesting if a sequence is heading in the same direction.
If it's any of those things.
A sequence a sub n is monotone
if that sequence is increasing or non-increasing or decreasing or
non-decreasing. So monotone amounts to jus a fancy way of
saying. Heading in the same direction.
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