Hey guys.
Welcome back.
Let's begin with our
part one. So the topics of
part 1 is sequences, limits
and completeness of R.
So in this part we'll
talk about what is a
sequence and what
is the limit of a sequence.
And what we mean by saying that R is complete.
Let's start with sequences.
I want to recall a
notation. So this funny
R. Funny symbol R is
the set of real numbers.
So it is a set of real numbers and real numbers, you can imagine
It is the collection; a real number is a point that is on the real line.
So you can see this is the
real line and on the right
end is positive infinity
on the left end is negative
infinity, and we collect all points that is between negative infinity and positive infinity.
And this funny
notation R is the set, the
collection of all the
points in R. All right.
So let's look at our first definition. This definition is talking about what is a sequence.
So a sequence. Wait a second. A sequence in R is an infinite ordered list of numbers
infinite oldered list of numbers in R.
So this is the definition of a sequence.
And normally we denote a sequence by a_n, parentheses n from 1 to infinity, or simply parentheses a_n.
And because this is the first definition in this course
So normally in a definition, when you see something that is in bold face like this sequence in here then
this is the terminology or the mathematical object that is defined in this definition.
So in this definition, "sequence" is the new terminology that is introduced.
And the meaning of this new terminology is the
It is an infinite ordered list of numbers in R. And also notations will also be in boldface.
All right.
So let's look at two examples of a sequence. So the first example is, if we consider this list: ordered
list of numbers -1, 1, -1, 1, -1, 1, -1, 1, and so on and just goes on.
And then this is a sequence in R, because it is an infinite ordered list of numbers.
So notice that actually in sequences, we allow repeated values to show up. In this sequence,
Uh a lot of repeated values.
Basically there are only two values: -1 or 1, and, for odd terms,  we got -1 and, for even terms,
we got 1's. And this is a sequence.
And sometimes when the sequence has some kind of special rule according to "n", then we can write it
compactly. So for this sequence, actually we can write it as (-1)^n, n from 1 to infinity
because (-1)^n, when you take n as integers 1, 2, 3, 4, then it just follows the same
rules as the sequence that we wrote here.
All right.
So this is an example of a sequence.
The second example is here.
So 1/2, 1/4, 1/8, 1/16, and so on.
This is a sequence in R, of course. It is an ordered list of real numbers.
And also it follows some rule. The rule is (1/2)^n.
So we can actually write it compactly as (1/2)^n,  for the sequence.
All right.
So this is uh examples.
These are examples of sequences. You can build your example. You just have to write a lot of numbers
and list them. If you give it some rules, that will be better so that you can write it in a compact form.
Okay so next I want to talk about what we mean by the convergence of a sequence.
So since we want to talk about the convergence of a sequence, we first have to give a sequence. So let
a_n, n from 1 to infinity be a sequence in R, and also because we want to talk about the convergence.
We want to talk about where the sequence converges to.
So we have to also have a number.
So a in R is a real number. And we say that this sequence a_n converges to a.
So this is the term that we want to define, the concept of convergence.
We say that a sequence a_n converges to a, as n goes to infinity
if the following is true.
So this is a logical statement.
So let me explain.
There are some terminologies that, I mean there are some symbols that you may not see before.
Let me explain.
So here's a symbol.
Here's another symbol.
Here's a
abbreviation.
So this upside down A is
actually a symbol that is usually used in logic.
It is called "for all" or some some people would say it's "for any".
But I will just stick with "for all". And this funny upside, it's not upside down, it is actually left and
right opposite E.
So this E is actually meaning that "there exists".
All right.
And here's an abbreviation.
s. and t.
And in this course s. and t.
"s dot t dot" will mean "such that"
so yeah.
These are the two, three; two symbols and one abbreviation that I want to introduce.
It is normally used in a lot of courses, especially math courses.
So yeah it's better to tell you this here.
All right.
Okay.
So let's look at this logical statement.
This logical statement is basically: for all epsilon greater than zero,
there exists N_epsilon in, oh by the way, here is another symbol.
So this is a funny N. Uh maybe I'd erase it.
So this funny N is kind of similar to the funny R here. The funny R denotes the set
of real numbers.
And this funny N denotes the set of natural numbers. By natural numbers, in here, we will mean positive
integers.
So it's actually the set of positive integers, 1, 2, 3, 4, 5, and so on.
So this funny N is this set.
All right.
So this logical is saying that
for all epsilon greater than zero,
there exists an N_epsilon. N_epsilon means that it's something that depends on epsilon, and this number
is in the natural numbers such that the absolute value of a_n minus a will be less than or equal to epsilon
for n greater than or equal to N_epsilon.
Okay.
So this may look a little bit mysterious at first glance.
So let me talk about the idea of convergence.
So here is the idea of convergence.
The idea is this.
We want that, as n goes up to infinity,
the sequence a_n, the term a_n, will be arbitrarily close to the number a.
This is what we mean by convergence right.
Intuitively, when n goes up, when n is very large, then the a_n should be very close to a, and
if n goes up to infinity, then a_n can go, can be arbitrarily close to a.
So how do we write it,
write this down concretely.
Let us interpret this idea in other words.
So in another.
In other words, no matter how small an error is given.
So given an arbitrary error, there should be a sufficient index, sufficient index is the
N_epsilon.
So if n is sufficiently large, sufficiently large means that there is a sufficient index such that the
if n is greater than this sufficient index, then it's called "sufficiently large".
So if we are given an arbitrary error epsilon greater than zero, then there is a sufficient index N_epsilon,
such that, when n is greater than or equal to N_epsilon, which means it is sufficiently
large, then the difference between a_n and a will be smaller than the error.
So the difference between a_n and a is measured by |a_n - a|. This is the distance between
the a_n and a. so yeah.
Let's look at this sentence again. No matter how small we are given an error epsilon greater than zero
we can find a sufficient index N_epsilon, such that, when n is greater than N_epsilon, the difference
between a_n and a should be less than that error.
So this is a concrete interpretation of how we should look at the limit or we should talk about the
concept of convergence.
So go back to the logical statement here.
So logic statement is basically what we wrote in the second dot,
bullet point here.
So, for any epsilon greater than zero, which represents the error, for any error greater than zero then we can
find a sufficient index, such that, if the n, the index n, is greater than the sufficient index, we
actually has a_n and a in-between, distance between a_n and a is less than or equal to the pre-assigned
error.
So this is what we mean by convergence.
So we say that a_n converges to a, as n goes to infinity,
if this logical statement is true. Notice that here's a "for all epsilon greater than zero".
So we have to check for every epsilon, whether this kind of  N_epsilon exists.
If this is the case, then we say that a_n goes to a, as n goes to infinity.
All right.
So yeah. Because these words "a_n converges to a, as n goes to infinity" is too long,
sometimes, because mathematicians like symbols, symbols are easier to write down, so most of the time
we'll write limit, n goes to infinity a_n, is equal to a, which is equivalent to a_n converges to a
as n goes to infinity.
Okay.
And here's another terminology, called "convergent".
This is an adjective because "converge" is a verb.
So this is a verb.
And this is an adjective.
All right.
So sometimes we want to use adjective, and we say that a sequence is convergent if it converges to
somewhere. That means, if there is a real number a, such that, and a_n goes to a, as n goes to infinity.
Otherwise, we say that the sequence is divergent, so divergent is basically opposite of convergent. Actually
the negative, the negative of "convergent".
Negation of "convergent".
And sometimes we use the phrase "the limit of a_n exists". By saying that,
a_n converges to somewhere, there exists an "a" such that a_n converges to a.
And also we call the number "a", the limit of the sequence.
So when a_n goes to a, as n goes to infinity, we say that a is the limit of the sequence.
So notice that, in this definition, we define the concept of convergence, a_n converges to a,
as n goes to infinity,
if the following logical statements is correct, is true, and also introduce a notation "limit, n goes to infinity, a_n"
is equal to a.
This is a notation, and also define the adjective convergent and divergent then also the phrase
"the limit of a_n exists" and also we define the limit of a sequence.
Yeah the limit of a convergent sequence.
Yeah if a_n is convergent, then the place where the sequence converges to is called the limit of
the limit of the sequence.
All right.
Um yeah.
There is another point I want to mention, because, uh, the notation N_epsilon is sometimes re-
a little bit redundant. it's nuh
It's kind of necessary to indicate that epsilon plays a role in N_epsilon, because of different epsilon
then you may have different N_epsilon. But sometimes people were saying, add the epsilon, the subscript
epsilon is kind of redundant.
So we just write simply write N.
So next time, or somewhere else, if you see N, then you have to keep in mind that this N actually depends
on epsilon.
okay.
So, let's now look at one example. We want to prove that, we want to show that the sequence 1/n
goes to 0, as n goes to infinity, or equivalently, uh, as we write in notation here, limit, n goes to infinity,
1/n is equal to zero.
Okay.
So just a reminder, this is the same as saying that, this sequence (1/n), n from 1 to infinity.
This sequence goes to 0 and goes to infinity.
All right.
So actually we have to follow the logic statement here.
And because this is on the previous slide, I will rewrite it here.
So, limit, n goes to infinity, 1/n, equal to zero
is actually equivalent to, for all epsilon greater than zero,
there exists an
N, depending on epsilon, which is a natural number, such that,
|1/n - 0| is less than or equal to epsilon, for all n greater than or equal to N. Just a reminder
that this N actually depends on epsilon. All right.
Um.
OK.
So this is what we want to prove.
So want to prove this statement. Here.
So you can see here
OK.
So yeah, this statement, to prove this statement, basically we have to check, for an arbitrary given epsilon,
whether we can find a corresponding N depending on epsilon, such that this statement is correct
|1/n - 0| less than or equal to epsilon, for n greater than or equal to N.
So to do this, in mathematics, we actually, the first step is to let an epsilon greater than 0.
So if we let this epsilon greater than 0, because we don't specify what epsilon is, for we don't specify
epsilon is equal to 1 or equal to 2 or equal to 1/2, and so on.
So the following statement will work for an arbitrary epsilon.
So if we let an arbitrary epsilon greater than zero and also find out a corresponding N_epsilon
without really using any kind of specific number for epsilon, then actually this argument will
work.
So if we let an epsilon greater than zero, and also find an N_epsilon such that |1/n - 0|
is less than or equal to epsilon, for n greater than, uh yeah, for all n greater than or equal to N_epsilon
Then actually we are done.
So the first step is let an epsilon greater than zero, and try to achieve this. If we achieve this, then we are basically
done, because epsilon is arbitrary, then we really prove that, for epsilon greater than zero, there exists
N, such that, |1/n - 0| is less than or equal to epsilon, for all n greater than or equal to N.
All right.
Um.
And actually you can notice that 1/n minus 0 with absolute value is the same as 1/n
because 1/n is positive and 0 is 0. 1/n - 0 is 1/n. Its absolute value is 1/n.
So we can simplify this term as 1/n less than or equal to epsilon.
So.
So actually this is our goal. We have to find an N_epsilon such that this is satisfied.
All right.
So here it is very simple to prove. The key idea is that, we can let N_epsilon equal to one over
epsilon with the ceiling function.
Okay.
So there are two things I want to explain.
The first thing is this notation.
So here is the two dots and equal sign (in fact, symbol)
In mathematics this means "defined as"
Yeah, because there are two dots and dots
starts with a d.
So it's "defined as"; defined equal to this.
So basically we defined the N_epsilon, as the ceiling function of one over
epsilon.
What is the ceiling function?
The ceiling function is the smallest integer.
So.
OK.
The ceiling function. The definition is here.
If we put a number, inside the ceiling function, x
Okay, so the ceiling function x, ceiling function of x, is the smallest integer n_x, such that, n_x is greater
than or equal to x.
So, for example, like uh, if we take the ceiling function of 1/2, then this will be equal to 1 because
1 is the smallest integer that is greater than or equal to 1/2.
And also like you would take like 5/2 ceiling function, then because 5/2 is 2.5 and 3 will
be the smallest integer that is greater than or equal to 5/2.
And this is the same for negative numbers.
So like if it is -5.6, then this will be equal to -5, because
-5 is greater than -5.6 and is also the smallest integer that satisfies this condition.
All right, so this is ceiling function.
The reason why we take ceiling function is because we then have this property because we know that
ceiling function of x is always greater than or equal to x.
Yeah, ceiling function only has this
Let me rewrite it. Hold on.
So the ceiling function of x is always greater than or equal to x.
This is always true by definition.
So because we set N_epsilon equal to 1 over epsilon ceiling function, we must have
N_epsilon greater than or equal to 1 over epsilon, which is this inequality, or we can take the reciprocal
on both sides, so that we get 1/N_epsilon less than or equal to epsilon, here.
So what is the benefit of this? If we have this equality then actually, if n is greater than or
equal to this N_epsilon, then we, of course, have 1/n less than or equal to N_epsilon, because we take
the reciprocal for this one. And because 1/N_epsilon is less than or equal to epsilon, so we also
have the second inequality.
This makes that 1/n is less than or equal to epsilon, for any n that is greater than
equal to N_epsilon,
which is what we want. Recall that this is what we want.
Right.
So this is wanted
by taking N_epsilon greater than or equal to
one over
epsilon,
we actually achieve the wanted statement and this completes the proof.
Proof complete.
So the key idea is just simply take N_epsilon equal to 1/epsilon ceiling function, then
the following arguments were automatically follow.
