Last time, then, we introduced the extended real line,
which was the real line
with two extra points,
one at minus infinity and one at plus infinity.
And we introduced our new total order on that space.
And I explained how to put
a metric on this space so that you could define
notions of convergence. We haven't said very much about
the convergence you get.
But the idea was that we would extend,
we're trying to extend the usual notion of convergence
from the real line to give you
some new convergent sequences which
converge to plus or minus infinity.
And that's where we got to last time.
So how ... so we've got
the usual convergent sequences
any 'old' convergent sequence of
real numbers that you already had before,
still converges, but we've got some new ones.
As I mentioned at the very beginning of last time,
if x_n = n, then x_n tends to infinity as n tends
to infinity, is one of my
favourite statements, because this says that
n tends to infinity as n tends to infinity. Okay!
But this should be interpreted - the second 'as n tends to infinity'
is, this second
'as n tends to infinity' is
just our notation, because we are talking about convergent sequences.
And the first one, which is also n tending to infinity,
is the statement, is
the sequence in the space converging to a point in the space,
because infinity is now in the non-negative extended real numbers.
So it sounds like a tautology,
but there is actually some content.
So what, how do you say what it means
for, that a sequence tends to infinity as n tends to infinity?
So let x_n be a sequence in the non-negative extended real numbers.
Well, you could go to the metric,
but if you don't want to use the metric,
here's another way of doing it:
x_n tends to infinity if and only if
oh, OK, as n tends to infinity ...
can anyone suggest something,
using some quantifiers? And for all
positive real numbers N, and things like that?
Anybody want to have a go?
[Student states definition] Yeah, that's exactly right!
You have to say that, for every positive number,
no matter how big, you eventually get past
that and stay past that when you're
in the sequence.. I usually use capital M rather than capital L,
but that doesn't matter. Your capital L is absolutely fine.
I'll use a capital M, just because I normally
do. For all M, and I had better
be careful when I say greater
than 0, because I don't want to allow infinity!
so I'll just make it absolutely clear that
for all M real numbers,
(we can't allow infinity, because that would give you something daft)
there exists a natural number N big enough so that,
(and then the last bit is the bit that
says that if you go past that point,
then you've got past M)
for all n in N with n greater or equal to capital N,
we have x_n has to be greater or equal to capital M.
It doesn't matter whether you use greater
or equal or greater than there:
let me rewrite that
that doesn't matter whether you use greater than or greater or equal.
If you actually go back and interpret
it in terms of open balls,
you may find that it's more natural to use greater than
there. I have a tendency to use
greater or equal, it doesn't make any difference.
They are equivalent definitions.
There's a useful little Sandwich Theorem you can use,
well the Sandwich Theorem works,
(squeeze rule) works in [0,infinity].
In particular,
Sandwich Theorem,
you want to find two sequences,
squeezing another sequence between them.
And if the two sequences tend to the same thing , then
the one in between will have to tend to that thing as well.
Now every sequence is squeezed from above by plus infinity.
So if you have a sequence below you tending to plus infinity,
the squeezing from above by plus infinity
is automatic and you must converge to plus infinity.
So if (x_n) is contained in [0,infinity],
and (y_n) as well
actually, and x_n less than or equal y_n
for all n and,
x_n tends to infinity,
then y_n tends to infinity as n tends to infinity as well.
It's got nowhere else to go!
If y_n is greater or equal x_n and x_n is heading off to plus infinity
y_n is just further to the right.
You can do the same thing
if you're working in R bar, you can do the
same thing, and you can work with minus infinity as well.
Similar definitions for x_n
tending to minus infinity.
We won't work with minus infinity very much in this module.
But if you go back and look at measure and integration,
you'll find minus infinity is important as well,
so you need to go back and
do, just imitate exactly what we just did,
but think about minus infinity.
And now again,
now I mentioned last time that if you're,
that every set in the extended real line
has got a supremum and an infimum,
a least upper bound and a
greatest lower bound in
the extended real line.
You didn't have to check for non-emptiness,
though the empty set is silly, but it's still true.
And you didn't have to check for bounded above or bounded below,
because you've already got at least one upper bound, plus infinity,
at least one lower bound, minus infinity.
So that meant everything had an inf and a sup.
Now similarly, when you're working in the real line,
you used to have a theorem, a Monotone Sequence Theorem,
which had a slightly complicated statement.
Monotone sequences, I'll remind you what those are in a minute,
but basically they're either sequences
where either none of the term ...
either every term is greater or equal to
the previous or, alternatively, every term
is less than or equal the previous. They're the
two kinds of monotone sequences; you're not allowed to have
some going up and some going down.
Monotone sequences in the real line can do
three things: they can converge to a real number,
they can diverge off to plus infinity, or they can diverge off to
minus infinity. In the extended real line,
then, that becomes much simpler: in the extended real line,
every monotone sequence converges,
whether it's to a real number, plus infinity or minus infinity.
So the statement becomes simplified.
Every monotone sequence in R bar converges in R bar.
I'll just remind you,
we use the terms nondecreasing sequences
and nonincreasing sequences.
For a nondecreasing sequence, this would be like x_1
less than or equal to x_2 less than or equal to x_3, and so on.
And a nonincreasing sequence,
that's x_1 greater or equal to x_2
greater or equal to x_3, greater or equal to.
You can use the terms 'increasing sequence' and 'decreasing sequence'
but ... for these, but then you have to,
you may want to distinguish between
strictly increasing and strictly decreasing sequences,
and supposedly these are the unambiguous terms,
though to be honest, I think that no matter.
how you say it, there's always an, there's always
something to worry about. But what you have to do
is you have to just see what
definition I'm working with here.
But these are the two kind
of monotone sequences.
Now, you're in R bar,
so, I mean, you
could also think of this as monotone ...
nondecreasing you can also
call
monotone, monotone increasing if you want,
as long as you distinguish from
strictly increasing, and nonincreasing,
a decreasing sequence, as long as
remember it doesn't mean strictly decreasing,
doesn't have to. And either
type, both types, by this Monotone Sequence Theorem,
which you can check the details of,
must converge in R bar. Either type, though, could
converge to any limit you like in R bar depending on what you want.
And, though some of them are tough to do.
For example, a nondecreasing sequence ...
How can you arrange for a
nondecreasing sequence to tend to minus infinity?
There's actually only one way to do it.
[Student suggests answer] That's right!
That's the only way to have a nondecreasing sequence
tending to minus infinity, is to be minus infinity forever.
Because if you're nondecreasing,
if you ever leave minus infinity,
you can never get back again,
and you can't even get close.
So as soon as you leave minus infinity,
you've had it! So if a nondecreasing sequence is to tend to
minus infinity, it's got to stay at minus infinity forever.
Similarly, if a nonincreasing sequence is
to converge to plus infinity,
it has to start at plus infinity and stay
there forever, that's the only way. Every other limit,
there's lots of different ways of doing it. So
if you want some real number,
you could have a sequence that has some infinite terms in
it, and then eventually becomes a sequence of real numbers
converging to that real number, that's what happens. If you
want a sequence of extended real numbers to converge to a real number,
you'll have finitely many terms that are plus or minus infinity,
and then after that you'll, you have to have real numbers.
And then it turns into an ordinary real sequence
converging to the real number.
So, but there are quite a lot
of ways for nondecreasing sequences to tend to plus infinity,
because that could be done,
well, I've already given you that
n tends to infinity as n tends to infinity.
So, so I'll comment here then
our, the only nondecreasing sequence
tending to minus infinity in R bar is the constant sequence ...
that's the only nondecreasing one, of course,
there's plenty of other ones.
There are plenty tending to plus infinity.
So sequences of real numbers,
can tend to plus or minus infinity,
(that's any old sequence) tend to plus or minus infinity.
But nondecreasing sequences of real numbers
can't tend to minus infinity.
We already saw that n tends to infinity
as n tends to infinity, I'll just remind you of that.
and sequences in R bar which
converge to real numbers are closely related to real sequences
converging to real numbers. Well, I won't write that down,
I've said that already.
You can think about that one. Okay!
I think that finishes Section 3.1.
Does anyone have any questions on the extended real line so far,
and convergence in the extended real line?
Then if not, I'm going to stop the recording there.
I'll start a brand new recording
for Section 3.2.