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Let's control the size of sequences.

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[SOUND]

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I want some terminology, some language for
us to be able to

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talk about sequences that don't get too
big or don't get too

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negative.
The word that

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we'll be using is bounded.
Let me give a precise definition.

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Precisely, to say that a sequence is
bounded above means that there's some real

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number m that's the bound.
So that for any index that term in the

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sequence is no bigger than m.
We can think about this graphically.

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Well, here's the graph of some sequence.

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Each of these dots represents a term in
the sequence, and

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the height of the dot represents the value
of that particular term.

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And I've positioned these terms

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in sort of a compressed way.
You'll notice that my labels here on

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the n axis.
Aren't all equally spaced, right?

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I'm squishing them together over here so I
can fit the

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entire future of this sequence on a single
sheet of paper.

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Now the sequence is bounded, and what that

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really means is that these terms never
exceed some

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bounding value.
Here I've picked some value m.

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No term of the sequence is above this
horizontal line.

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This horizontal line is representing an
upward bound for this sequence.

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I can also bound a sequence from below,

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meaning that the sequence doesn't become
too negative.

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To say that a sequence is bounded below.

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Means that there's some real number m.

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That's the bound.

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So that for any index n, the nth term

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in the sequence is no smaller than that
bound m.

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Sometimes a sequence will be bounded both
from above and from below.

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Well in that case, we just say the
sequence is bounded.

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So a sequence is bounded exactly when it's
bounded both above and below.

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Now, it's worth pointing out that the
definition

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for bounded below involves a number M.

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And the definition for bounded above
includes a number m.

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But it's very unlikely that the upper
bound and the lower bound are the same.

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of course they could be the same, right,
if the sequence is just the constant

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sequence, but in general bounded sequence
is going to

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have a different upper bound and lower
bound.

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So we've got our precise definitions.

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Now we'll have to make up some sequences
and

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try to figure out whether those sequences
are bounded.

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For example, let's think about this
sequence.

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This sequence ace of n is sine of n.

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This sequence is suspended above, bounded
below, bounded?

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Neither bounded above nor below?
Well let's think about it.

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What do we know about sine?

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Sine, no matter what I plug into sine,
it's no

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bigger than one, and it's no smaller than
minus one.

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Consequently, the sequence is bounded
above by

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one and bounded below by minus one.

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This is an upper bound, this is a lower
bound, for that sequence.

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And since the sequence is bounded both
above and

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below, I can just write that the sequence,
is bounded.

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you don't want to get the idea that this
one and

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this minus one are the only choices for
upper and lower bounds.

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I mean I

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could also say that the sequence is
bounded above by 17.

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That would be accurate to say.

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so there's lots of choices here, for this
upper bound,

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but in any case, in this example, the
sequence is bounded.

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The sequence sine of n is bounded.
Let's do another example.

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So, let's look at a sequence, b sub n is n
times sine of pi times n over two.

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Is this sequence, boundary above, boundary

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below, bounded, neither bounded above nor
below?

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How can we think about his?

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Well, this is sine of pi times a whole
number over two.

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What are the possible values for this sine
term?

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Well this thing could be zero, it could
also be one, it could also be minus one.

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And they'll be very large inputs, very
large n for

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which sine of pi times n over two is one.

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And for those very large inputs, b sub n
is just n times one.

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There'll be other very large inputs for
which sine of pi times n over two is

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negative one, and for those very large
inputs, b sub n is going to be negative n.

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So that means that this sequence isn't
bounded above and it isn't bounded below.

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It's neither bounded above nor below.
Now I can write out more formal argument.

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Let me

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try to write out a formal argument that
this sequence is not, bounded above.

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But to make that argument precise, I'm
going to try to convince you that it's

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not bounded above by m, but I'm not
going to tell you what m is.

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So I'm going to try to write down an

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argument that shows it can't be bounded
above by

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m, and since it can't be bounded above

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by an arbitrary m, it's just not bounded
above.

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There is no upper bound.

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So, how do I know that this sequence above
is bounded by m?

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Well, the trick is to pick some, some
other number.

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I'm going to pick some number that looks
like this.

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One followed by a whole bunch of zeros
followed by a one.

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But, I want to pick that number, so that
it's bigger.

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Than your purported upper bound.

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And let's call that number big n.

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So no matter what m you pick, I can find a
whole number like this, one followed

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by a bunch of zeros followed by a one
which is bigger than m.

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And what do I know about b sub big n?

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Well this is n times, sine of pi times big
n over two.

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But if you think about it a bit, sine of a
number like this times pi over two is

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one.
So for this particular value of big n,

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b sub n is just n times one is just n.

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So what do I know about big n?
Big n is bigger then m so that

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means that b sub n is bigger then m but

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that means that m can't be a upper,

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bound.

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Big m can't be the big m that appears in
the definition of bounded above.

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And because this sequence isn't bounded
above by an arbitrary big m.

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It's just not bounded above.

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There's no choice of m for which the
sequence

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can be said to be bounded above by m.

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That might seem too formal.

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Let's try and do some numeric
calculations,

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just to get a sense of what's going on
here.

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Numerically, what's going on here?

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Well, b sub one is one times sine of pi
over two.

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That's just one.

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B sub two is two times sine of two times

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pi over two, which is sin of pi, which is
zero.

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It's just zero.

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B sub three is three times sine or pi
times three over two.

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The sine of pi times three over two is
minus one, so b sub three is minus three.

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Well how do the rest of the terms go?

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Let me just start writing them out.

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So one is the first term.
Zero is the next term.

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Minus three is the next term.
If I plug in four, for n, I get zero.

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If I plug in five, I get five.

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If I plug in six, I get zero.
If I plug in seven, I get minus seven.

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If I plug in eight, I get zero.
If I plug in nine, I just get nine.

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If I plug in 10, I get zero.

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If I plug in 11, I get minus 11, and it
keeps on going.

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Every other term is zero, and the

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non-zero terms are flip-flopping in sign,
in s-i-g-n.

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So one, five, nine, and the next term 13,
are all positive.

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But negative three,

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negative seven, negative 11 and so on,
these are, these terms are negative.

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So this sequence isn't bounded above and
it isn't sounded below.

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If I go far enough out in the sequence, I
can find terms that are very positive.

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And if I go far enough out in the

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sequence, I can find terms that are very
negative.

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[SOUND].