Okay, that is interesting let us probably
say that also. Now, a n is a sequence which
is given to you a 1, a 2, a 3 and so on, a
n so on. Here is something which happens very
often, you are standing in a queue, and suddenly
somebody with a, some tag or something comes
“tum bahar aajao, tum bahar aajao, tum bahar
aajao, special queue tumhare liye bana diya,
you are VIP’s”, a separate queue. So,
what you are doing?
From that ordered collection, you are picking
out some members, but he is not allowed that
after he has picked up the tenth person, he
can go back and look “koi aur bhi bacha
hai ke nahi, woh nahi allow karenge, ek bar
tumne nikaal liya toh nikaal liya”, go ahead
and pick up whichever you want. So, you are
allowed to pick up elements of a sequence
in a increasing order, whichever you like.
For example, you can pick up this one, now
go on, you can pick up this one go on, you
can pick up this one. So, this one you can
call it as n 1, your selection, first selection,
keep the hierarchy in the queue, do not come
back, pick up the second one. This is a n
second selection, third selection a n 3 and
so on.
So, in general a n k you can form a sequence
by picking up elements from the given sequence,
but keeping in mind n 1 should be strictly
less than n 2, n 2 should be strictly less
than n 3, and go on, is it okay? So, such
a thing is called a sub sequence. So, this
is called a subsequence.
Where n 1 is less than n 2 less than and so
on. So, what I am doing? So, pick up a strictly,
monotonically increasing sequence of numbers,
n 1, n 2, n 3 and so on. According to that
pick up the elements of the given sequence
that gives you a sub-sequence, is it okay?
How to form a sub sequence? Now, if a sequence
is convergent, what can you say about the
subsequence?
Obviously it should be, because if those a
n’s are coming closer to a value L, part
of that also should come inside, okay? So,
first fact try to write a proof yourself if
you like, okay, that if a sequence a n is
convergent that implies every sub sequence
must also converge to the same limit.
Can I say converse is true? Of course, if
you will look at the sequence minus 1 to the
power n, I pick up the odd terms, they converge
to plus 1, even term converge to minus 1.
But supposing I put a condition every sub
sequence has to converge to the same limit
then it is obvious it should be happen that
your sequence itself should converge to that
same value.
So, this is a theorem again we will not prove
it, we will not ask you in the exam, but it
is a simple obvious fact intuitively quite
clear, a sequence converges to a limit L if
and only if every subsequence converges to
the same limit namely L.
So, with that intuitive understanding let
me go over and something. Okay, here is something,
so let us assume a n 
is a sequence, so we have just now observed,
a n convergent if and only if every subsequence
converges to the same value, is it quite interesting,
try to think of proof yourself, you will see
how logic is playing a role there. Here is
something which all of you should understand.
If you want to understand when a statement
is true, you should understand when the statement
is false, this is a fact of mathematics and
of life, if you want to know what is true,
it is as good as knowing what is false. If
you do not know what is false, you will not
know what is true, think about it. And the
logic is played in mathematics also, if you
want to say some statement, okay.
See, okay, I think let me elaborate that before
writing it somewhere here, where do I write
it let me write it here, a n converges to
L, that is a statement, that is same as saying
for there exists some L, I think we did that
last time, L belonging to R such that a n
minus L is less than epsilon for every n bigger
than n naught, that is convergence.
So, truth of the statement that a n is convergent
is equivalent to writing this that there is
a number L which is going to be the limit
and what does that mean? For every epsilon
there should be a stage after which a n should
come closer to. If this statement is not true,
what does it mean?
So, there is a now the other way around now,
this is, if you want to understand what is
true, you should understand what is false.
So, what is the falsehood of this statement?
a n is not convergent, that means what? The
first thing was there exist some number L,
so, this should go bad.
That means what? That means for every L, you
may have problem later on using this slide,
because something is coming somewhere. So,
let me I think let me not do that let me go
over to here. So, a n converges to L is equivalent
to saying there exists L belonging to R such
that such that R and there exists some n naught
belonging to n such that mod of a n minus
L is less than epsilon for every n bigger
than n naught.
And I want to say that if this statement is
not true, a n is not convergent. So, what
is that equivalent to? So, first of all this
statement should go bad. So, this is a combination
of many statements, first one is there exist
some L such that something happens, so, it
should go back that means, if it is not going
to be true for one particular L that means
for every L it should be bad.
So, for every L belonging to R, what should
happen? Here it says there exist a stage I
should not be able to find this stage, that
means for every n whatever I think n is the
stage, there exists some stage after that.
So, let us call it as n, n 1 bigger than n
such that this is, this goes bad that means
mod of a n 1 minus L is bigger than or equal
to epsilon. Whatever, there is no number that
means for every number. And what happened
to that epsilon wherever what happened to
the epsilon I did not write that?
For such that for every epsilon was there.
Because epsilon will really not come into
picture. For every L given an epsilon, that
means what? There exists at least one epsilon.
So, there exists I should write there exist
some epsilon such that this goes bad. At least
there is one interval, so that you say that
after that see everything is inside? No, at
least one of them will go out and that is
what is we are saying.
So, this is what I am saying trying to understand
the negation of a given statement, a n is
conversion, so try to do it every time whenever
you find a theorem or something anywhere in
your understanding, even in our life.
How do you compare something is true? You
have to compare it with something which is
false. In probability theory, you will all
be doing probability theory. What is a chance
of this fan falling down just now? Or in a
particular date at a particular time? Chance
is probability. That is equivalent to knowing
the probability of this not falling down,
both are equivalent statements, one is the
truth that it will fall, other is the falseness
it will not fall.
So, mathematically also you will find these
things coming back to you because you are
all ASI students, that is same. Okay, so,
let me not spend much time on it and go back
to what I was saying is corollary we approved.
So, I said every subsequence converges for
a given sequence.
I think the more things let me right here
now. So, a n is a sequence okay, so, one we
have already said, every sub sequence converges
if and only if the sequence converges. Now,
suppose a n is convergent, why limit is not
going to be important, so, a n is convergent,
again convergence means, elements are going
to come closer to it.
Let us not bother about to what it comes closer
let us see the effect of this, if L is the
limit it should come inside this, I can make
it smaller, is for every epsilon still the
a tail should come inside, still a tail should
came inside, so, what should be happening
to the terms of the sequence itself?
If and if I do not know L the limit, but I
know it is convergent, that means the terms
of the given sequence must come closer to
each other as you progress, as n becomes larger
and larger a n should become closer and closer
to each other, because if they remain away,
they are not going to converge.
So, it looks like a intrinsic property of
the sequence convergence, a sequence convergence
should imply that the terms are going to come
closer to each other, because if I look at
the sequence whether I know the limit or not
does not matter, I can just say that the terms
are coming closer to you, I look at that person
and say he is honest, so that is an intrinsic
property, without verifying whether he has
a criminal case against him or not that is
an extrinsic property. Limit is something
outside which is not given to you, but saying
terms are coming closer is a property of the
sequence I can verify, so, let us give it
a name.
So, let us call a sequence, so, we say a sequence
a n is Cauchy, Cauchy was a mathematician
and his name I think you will find he will
come in your course also somewhere or the
else, in mathematics he comes left and right,
in mathematics courses and statistics also
he will come somewhere, okay.
We say is a Cauchy sequence if I can find
there exist some stage n naught, everything
is a property of the tail in sequences and
now it such that a n minus a m the distance
such that if I should write that Cauchy if
for every epsilon bigger than 0 the distance
is epsilon for every n bigger than. How close?
As close as you want it, but for a tail. Given
epsilon you want this close that will be a
stage, much closer you want some other stage
will be there, but this is a property of saying
it is Cauchy. And I said it is an intrinsic
property.
So, claim, here is a claim convergent, I should
write better, convergent implies Cauchy. Cauchy
as human being, convergent is a property,
looks very odd writing convergent implies
Cauchy, means every sequence which is now
we are using them as adjective, if a sequence
is Cauchy, convergent then it is also Cauchy.
So, we are not using it as a noun, we are
using it as a adjective, okay.
So, let us see how is it true? Now, here is
convergent means there is a L, there is L
minus epsilon and there is that is a neighborhood
that is a interval such that for every n bigger
than n naught, a n's are here, a n is here,
a m is here when n and m are bigger than n
naught what that obviously says the distance
between a n and a m is small.
So, let us write it mathematically. How do
I write it mathematically? It will like, we
can write it mathematically as a n converges
to L implies there exists some stage n 1 such
that or n 0 such that mod a n minus L is less
than epsilon for every n bigger than n naught.
What I want? a n minus a m, that is my target,
but I know something about a n minus L and
a m minus L.
So, let us bring in add and subtract, so less
than or equal to for every n bigger than this
is true for all L and m. So, in particular
for n and m bigger than or equal to n naught
and that says is less than 2 epsilon, because
each is less than epsilon, because of convergence,
is it okay? Because of this property each
is less than epsilon.
Now, this is a what I call at cosmetic surgery,
I do not want two epsilon, I want only epsilon.
To look it nicely, so, I will go back and
say it epsilon by 2, so it will be epsilon
by 2 by 2 that is epsilon. That is a minor
thing, because epsilon is arbitrary, so, I
can change it to anything I like to start
with okay.
So, that says every convergent sequence is
Cauchy. So, Cauchyness is a necessary condition
for a sequence to be convergent, like boundedness.
Question asked, now, can I say if a sequence
is if the terms are coming closer and closer,
will the sequence converge? For rationals,
it is not true, for reals it is true.
So, we want to prove a theorem that every
Cauchy sequence is also convergent. So, it
is a equivalent way of saying convergence,
sometimes you are not interested in knowing
the limit of the sequence, you are only interested
in knowing the property of the sequence, there
Cauchyness is very useful because you do not
have to bother about the limit you have to
only bother about, given the sequence, intrinsically
look at whether the terms are coming closer
or not.
So, we want to prove a statement that every
Cauchy sequence is also convergent. So, we
will prove that probably next lecture because
there are only 3 minutes and we cannot prove
it, but here is something I want you to sort
of have a look at it.
Given a sequence a n there is one way of visualizing
it, I want to visualize this sequence, not
as points on the line, but as if they are
heights of poles, a 1 is some height, a 2
is some height, so, a 1, a 2, a 3, a 4 may
be a 5 and such things. Imagine a sequence
to be saying the height of something, a building
probably or something.
So, when will you say a sequence is increasing?
Imagine if they are buildings for example,
you are able to see the top of this building,
increasingly you are able to see that top
of the next building also, this building does
not obstruct the view of the next one everything
is visible and you want to say it is monotonically
decreasing then everything below is visible.
If it is something like this, you cannot say
that this is increasing because something
is obstructed in between, but at least it
looks possible. Let us start with something
and let us see the next building which is
visible to you and the next building which
is visible to you and then the next.
If I pick up these buildings, what I will
get? I will get a sub sequence which is monotonically
increasing or sometimes I can see this and
then probably I forget about this building
then I can see if this building is not there
if this is not there, then I can see the next
one here which is smaller than this, probably
there is something smaller than this, then
if I (obs) if I do not forget about these
ones, then I can see that also. So, then I
will be picking up something a subsequence
is monotonically decreasing.
So, it seems to be a fact that every sequence
has either a monotonically increasing subsequence
or a monotonically decreasing subsequence
this picture seems to seems to says through
me that. So, we will prove that next time
more mathematically that every sequence has
got a subsequence, which is either monotonically
increasing or monotonically decreasing.
Now for a convergence, I need a bounded. Suppose,
I am given a bounded sequence then combine
it with this result now there is a subsequence,
which is monotonically increasing or decreasing
and the original sequence is bounded, then
this must be bounded. So, every sequence which
is bounded will have a convergent subsequence
that is again an important theorem is called
Bolzano Weierstrass property.
Now, look at a Cauchy sequence. To say that
a Cauchy sequence is convergent, it is enough
to prove a sub sequence is convergent, because
terms are coming closer, if a sub sequence
is converging, the sequence itself must be
coming closer to that value because they are
also coming closer to each other.
So, another fact that a Cauchy sequence is
convergent to prove this it is enough to produce
a subsequence which is convergent. Now, a
Cauchy sequence has to be bounded also because
they are coming closer and closer given epsilon
only finitely many can be outside.
Again does not matter where, again a Cauchy
sequence is going to be bounded, Cauchy sequence
is bounded, Cauchy sequence has got a monotonically
increasing subsequence, by the Bolzano Weierstrass
property it must converge. So, a Cauchy sequence
has got a convergent subsequence, hence the
sequence itself must converge that will proof
Cauchyness is equivalent to convergence. So,
I have already given you a trailer of the
next class, so, we will do it next time. Okay.
Thank you.
