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.