3. Sequences of Numbers
3.1. Sequences
So far we have introduced sets as well as the number systems that we will use in this text. Next, we will study sequences of numbers. Sequences are, basically, countably many numbers arranged in an order that may or may not exhibit certain patterns. Here is the formal definition of a sequence:Definition 3.3.1: Sequence  
A sequence of real numbers is a function f: N R. In other words, a sequence can be written as f(1), f(2), f(3), ..... Usually, we will denote such a sequence by the symbol , where a_{j} = f(j). 
We now want to describe what the longterm behavior, or pattern, of a sequence is, if any.
Definition 3.1.2: Convergence  
A sequence of real (or
complex) numbers is said to converge to a real (or complex) number
c if for every > 0
there is an integer N > 0 such that if j > N then
 a_{j}  c  <The number c is called the limit of the sequence and we sometimes write a_{j} c. If a sequence does not converge, then we say that it diverges. 
We are going to establish several properties of convergent sequences, most of which are probably familiar to you. Many proofs will use an ' argument' as in the proof of the next result. This type of argument is not easy to get used to, but it will appear again and again, so that you should try to get as familiar with it as you can.
Proposition 3.1.4: Convergent Sequences are Bounded  
Let be a convergent sequence. Then the sequence is bounded, and the limit is unique. 
Proposition 3.1.6: Algebra on Convergent Sequences  
Suppose and
are converging to
a and b, respectively.
Then

This theorem states exactly what you would expect to be true. The proof of it employs the standard trick of 'adding zero' and using the triangle inequality. Try to prove it on your own before looking it up.
Note that the fourth statement is no longer true for strict inequalities. In other words, there are convergent sequences with a_{n} < b_{n} for all n, but strict inequality is no longer true for their limits. Can you find an example ?
While we now know how to deal with convergent sequences, we still need an easy criteria that will tell us whether a sequence converges. The next proposition gives reasonable easy conditions, but will not tell us the actual limit of the convergent sequence.
First, recall the following definitions:
Definition 3.1.7: Monotonicity  
A sequence
is called monotone increasing if
a_{j + 1} a_{j} for all
j.
A sequence is called monotone decreasing if a_{j} a_{j + 1} for all j. 
 Monotone increasing:
 a_{j + 1} a_{j}
 a_{j + 1}  a_{j} 0
 a_{j + 1} / a_{j} 1, if a_{j} > 0
 Monotone decreasing:
 a_{j + 1} a_{j}
 a_{j + 1}  a_{j} 0
 a_{j + 1} / a_{j} 1, if a_{j} > 0
 If a sequence is
bounded above, then c = sup(x_{k}) is finite. Moreover,
given any > 0, there exists
at least one integer k such that
x_{k} > c  , as
illustrated in the picture.
 If a sequence is
bounded below, then c = inf(x_{k}) is finite. Moreover,
given any > 0, there
exists at least one integer k such that
x_{k} < c + ,
as illustrated in the picture.
Proposition 3.1.9: Monotone Sequences  
If is a monotone
increasing sequence that is bounded above, then the sequence must converge.
If is a monotone decreasing sequence that is bounded below, then the sequence must converge. 
Using this result it is often easy to prove convergence of a sequence just by showing that it is bounded and monotone. The downside is that this method will not reveal the actual limit, just prove that there is one.
Examples 3.1.10:  

Theorem 3.1.11: The Pinching Theorem  
Suppose {a_{j}} and {c_{j}} are
two convergent sequences such that
lim a_{j} = lim c_{j} = L. If a sequence
{b_{j}} has the property that
a_{j} b_{j} c_{j}for all j, then the sequence {b_{j}} converges and lim b_{j} = L. 
Example 3.1.12:  
Show that the sequence sin(n) / n and cos(n) / n both converge to zero. 