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A list of numbers might keep getting
bigger and bigger.

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[MUSIC]

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I'm want to precisely define increasing
for sequences.

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A sequence a sub n is increasing If,
whenever m is bigger

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than n, then the mth term is bigger than
the nth term.

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This is capturing the idea that the terms
are

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getting bigger as I go further out in the
sequence.

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Let's take a look at an example of an
increasing sequence.

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For example the sequence a sub n equals n
squared is an increasing sequence.

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I can write out the first few terms.

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The a sub one term is one and that's 4, 9,
16, right?

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These are just the perfect squares and the
claim is that this

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is an increasing sequence, that as I go
farther out in the sequence.

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The terms get bigger and to really
convince you of that formally, what I

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need to show is that whenever m is bigger
than

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n, then a sub m is bigger than a sub n.

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And indeed that's true because a sub m is
m squared and a sub n is n squared.

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And as long as these m and n are positive
this is true whenever this is true.

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We can also consider a sequence which is
decreasing.

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A sequence

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a sub n is decreasing if whenever m is
bigger than

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n then a sub m is less than a sub n.

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This is capturing the idea that the terms
are getting smaller.

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It's telling me that larger indices
correspond to smaller terms.

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So, as I go further out in the sequence,
the terms are getting smaller.

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It's easy to build examples of decreasing
sequences.

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Well, if, a sub-n is an increasing
sequence So an example

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of this would be an example we just saw an
example would be a sub n equals n squared.

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Well I've got an increasing sequence then
the sequence b sub n

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defined by just negating the terms of a
sub n is a decreasing.

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Sequence.
Alright.

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If the terms of a sub n are getting
larger, the

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terms of b sub n are getting more
negative, they're getting smaller.

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Why even bother with all of this?

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Well, basically, I just want to give
definitions for

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the kinds of qualitative features that we
might

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be interested in, might be interested in a

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sequence that's increasing or a sequence
that's decreasing.

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We might also be interested in a sequence
which maybe isn't

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necessarily getting bigger but at least it
isn't getting any smaller.

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Is nondecreasing if whenever m is bigger
than n, then this isn't the case.

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And to say that isn't the case means
instead

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of less than it's greater than or equal
to.

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So a non decreasing sequence is not
getting any smaller

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in the sense that future terms are at
least as large as previous terms.

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Here's an example of a non-decreasing
sequence.

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Sequence might start 1, 1, 2, 2, 3, 3, 4,

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4 and so on where it sort of repeats
itself.

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And this sequence isn't increasing because
later

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terms are not larger than earlier terms.

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two is not greater than two.
But the sequence is non decreasing, right?

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At least the terms aren't getting any
smaller.

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Not very surprisingly we can also define
non-increasing.

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A sequence is non-increasing if whenever m
is bigger than n

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then a sub m isn't bigger than a sub n.
And isn't bigger means less

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than equal or to.
So a non-increasing sequence is, well, not

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necessarily decreasing but at least its
not getting any larger.

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So thus far we've talked about

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increasing and decreasing, non-increasing
and non-decreasing.

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Generally, it's just interesting if a
sequence is heading in the same direction.

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If it's any of those things.

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A sequence a sub n is monotone

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if that sequence is increasing or
non-increasing or decreasing or

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non-decreasing.
So monotone amounts to jus a fancy way of

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saying.
Heading in the same direction.

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[NOISE]