This web page contains basic definitions that will be needed in our discussion of Sequences and Series. Some of this will be familiar to you, some is new. The point of this web page is to set out some terminology and background so that it is clear where we are starting from. For basic definitions of terms (most of which you will need to learn) see the glossary and the other pages here.
You should be familiar with the usual number systems of mathematics,
the natural numbers, the integers, the rationals, the reals, and the complex numbers,
and be comfortable with using the symbols
The normal set-theoretical language is very important for any
detailed discussion of the reals and the other number systems.
We shall use this with the usual notation, including: curly brackets
There are two competing conventions for
You should be familiar with the idea of an interval on the set of reals. These are sets defined in one of the following ways.
One of the most important functions on the real numbers that gets the subject of analysis going is the absolute value function. You should already have seen this, and you must be sure you understand its definition and can use it accurately in inequalities, etc.
Note that this definition is by cases. Therefore all statements using absolute value will be expected to be proved using arguments that split into several different cases. Absolute value will be discussed in detail later when we talk about the very important triangle inequality.
Another important function is the integer-part function.
This is written
A variation is
Another important and much-used piece of terminology is that of
decreasing functions. Again, we will have quite
a lot to say about these later on, but we will need the terminology from
You should at this point in your mathematical career have seen
the idea of rigorous mathematics where all the important theorems
are proved from axioms, or basic facts about the objects under consideration
that are taken for granted. For example, you may
have seen the idea of a group, where one of the axioms is that the operation
is associative. Examples of groups include the set of real
Most of the questions we shall be looking into in these web pages
can be seen as an investigation of the detailed mathematical structure
obvious idea that turns out
to be quite tricky to pin down. The set
From another point of view, the idea of limit is a very practical one and underpins all practical mathematics, especially that involving calculus. The notion of limit is really a theory of approximations - of approximations that can be made as good as we shall ever need. This is a really important idea also for applied mathematics and computational mathematics, as well as pure mathematics.