So let us see what we've already
known and what we are going to define here.
The thing that we've known is sequences.
Sequences is basically functions
of natural domain where we have
a set of natural numbers and for
each natural number we have some value.
Okay. That's our already established definition.
So we are going to move forward into
the case of functions from
real numbers to real numbers, right?
So we need to discuss firstly what
is the difference between the structure of
the real number set and a natural number set,
and actually this structural difference
is quite observable.
Assume that we have
our real numbers drawn
on a straight line and then we just
point out all our natural numbers
and consider for example neighborhood of the point three.
I'm going to stress this out.
So what an idea.
Yes, for the natural numbers the closest as we can get
is is either two or four.
Right. So the smallest distance in between
natural numbers from number three is one but
as real numbers as you can all understand
there is no smallest distance between
three and a 1000 numbers because we can
get as close to three as possible.
So that's basically the concept
of limit point of any set.
A point is called limit if we can
get infinitely close to it by it's elements of this set.
So the idea is very simple here.
We can discuss the limits.
We can consider the limit of
the function at a given point only
if this point is
a limit point of the functional domain. All right.
So the question here is,
why we've considered the same case for sequences for
example and it's a matter of fact quite easy because,
let me do some kind of infinity here.
Says somewhere in the very right there is an infinity.
Of what is the phrase
as close to infinity as possible means actually?
It basically means that we can look at some beam
starting from some finite number
M [inaudible] as infinity and then of just moves
this M shorter and closer to the right.
So basically the idea is that
any neighborhood of infinity means that we
can assume that we have some quality,
some condition or for example our sequence does
not get further from it's
limits starting from the point M,
rising from the number M and further away.
So for the natural numbers,
the only limit point is infinity
because we can get closer and closer to infinity,
but for real numbers the set of limit points up or
the numbers plus and minus infinity.
So that's basically the idea of
how transition from sequences to Functions.
Previously we were speaking only
about limits at infinity and now we can
speak about point-wise limits
and we need to properly define
it using already existent definitions for our sequences.