Let's control the size of sequences.
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I want some terminology, some language for us to be able to
talk about sequences that don't get too big or don't get too
negative. The word that
we'll be using is bounded. Let me give a precise definition.
Precisely, to say that a sequence is bounded above means that there's some real
number m that's the bound. So that for any index that term in the
sequence is no bigger than m. We can think about this graphically.
Well, here's the graph of some sequence.
Each of these dots represents a term in the sequence, and
the height of the dot represents the value of that particular term.
And I've positioned these terms
in sort of a compressed way. You'll notice that my labels here on
the n axis. Aren't all equally spaced, right?
I'm squishing them together over here so I can fit the
entire future of this sequence on a single sheet of paper.
Now the sequence is bounded, and what that
really means is that these terms never exceed some
bounding value. Here I've picked some value m.
No term of the sequence is above this horizontal line.
This horizontal line is representing an upward bound for this sequence.
I can also bound a sequence from below,
meaning that the sequence doesn't become too negative.
To say that a sequence is bounded below.
Means that there's some real number m.
That's the bound.
So that for any index n, the nth term
in the sequence is no smaller than that bound m.
Sometimes a sequence will be bounded both from above and from below.
Well in that case, we just say the sequence is bounded.
So a sequence is bounded exactly when it's bounded both above and below.
Now, it's worth pointing out that the definition
for bounded below involves a number M.
And the definition for bounded above includes a number m.
But it's very unlikely that the upper bound and the lower bound are the same.
of course they could be the same, right, if the sequence is just the constant
sequence, but in general bounded sequence is going to
have a different upper bound and lower bound.
So we've got our precise definitions.
Now we'll have to make up some sequences and
try to figure out whether those sequences are bounded.
For example, let's think about this sequence.
This sequence ace of n is sine of n.
This sequence is suspended above, bounded below, bounded?
Neither bounded above nor below? Well let's think about it.
What do we know about sine?
Sine, no matter what I plug into sine, it's no
bigger than one, and it's no smaller than minus one.
Consequently, the sequence is bounded above by
one and bounded below by minus one.
This is an upper bound, this is a lower bound, for that sequence.
And since the sequence is bounded both above and
below, I can just write that the sequence, is bounded.
you don't want to get the idea that this one and
this minus one are the only choices for upper and lower bounds.
I mean I
could also say that the sequence is bounded above by 17.
That would be accurate to say.
so there's lots of choices here, for this upper bound,
but in any case, in this example, the sequence is bounded.
The sequence sine of n is bounded. Let's do another example.
So, let's look at a sequence, b sub n is n times sine of pi times n over two.
Is this sequence, boundary above, boundary
below, bounded, neither bounded above nor below?
How can we think about his?
Well, this is sine of pi times a whole number over two.
What are the possible values for this sine term?
Well this thing could be zero, it could also be one, it could also be minus one.
And they'll be very large inputs, very large n for
which sine of pi times n over two is one.
And for those very large inputs, b sub n is just n times one.
There'll be other very large inputs for which sine of pi times n over two is
negative one, and for those very large inputs, b sub n is going to be negative n.
So that means that this sequence isn't bounded above and it isn't bounded below.
It's neither bounded above nor below. Now I can write out more formal argument.
Let me
try to write out a formal argument that this sequence is not, bounded above.
But to make that argument precise, I'm going to try to convince you that it's
not bounded above by m, but I'm not going to tell you what m is.
So I'm going to try to write down an
argument that shows it can't be bounded above by
m, and since it can't be bounded above
by an arbitrary m, it's just not bounded above.
There is no upper bound.
So, how do I know that this sequence above is bounded by m?
Well, the trick is to pick some, some other number.
I'm going to pick some number that looks like this.
One followed by a whole bunch of zeros followed by a one.
But, I want to pick that number, so that it's bigger.
Than your purported upper bound.
And let's call that number big n.
So no matter what m you pick, I can find a whole number like this, one followed
by a bunch of zeros followed by a one which is bigger than m.
And what do I know about b sub big n?
Well this is n times, sine of pi times big n over two.
But if you think about it a bit, sine of a number like this times pi over two is
one. So for this particular value of big n,
b sub n is just n times one is just n.
So what do I know about big n? Big n is bigger then m so that
means that b sub n is bigger then m but
that means that m can't be a upper,
bound.
Big m can't be the big m that appears in the definition of bounded above.
And because this sequence isn't bounded above by an arbitrary big m.
It's just not bounded above.
There's no choice of m for which the sequence
can be said to be bounded above by m.
That might seem too formal.
Let's try and do some numeric calculations,
just to get a sense of what's going on here.
Numerically, what's going on here?
Well, b sub one is one times sine of pi over two.
That's just one.
B sub two is two times sine of two times
pi over two, which is sin of pi, which is zero.
It's just zero.
B sub three is three times sine or pi times three over two.
The sine of pi times three over two is minus one, so b sub three is minus three.
Well how do the rest of the terms go?
Let me just start writing them out.
So one is the first term. Zero is the next term.
Minus three is the next term. If I plug in four, for n, I get zero.
If I plug in five, I get five.
If I plug in six, I get zero. If I plug in seven, I get minus seven.
If I plug in eight, I get zero. If I plug in nine, I just get nine.
If I plug in 10, I get zero.
If I plug in 11, I get minus 11, and it keeps on going.
Every other term is zero, and the
non-zero terms are flip-flopping in sign, in s-i-g-n.
So one, five, nine, and the next term 13, are all positive.
But negative three,
negative seven, negative 11 and so on, these are, these terms are negative.
So this sequence isn't bounded above and it isn't sounded below.
If I go far enough out in the sequence, I can find terms that are very positive.
And if I go far enough out in the
sequence, I can find terms that are very negative.
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