This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e., the concept of a limit. Though Newton and Leibniz discovered the calculus with its tangent lines described as limits of secant lines, and though the Greeks were already estimating areas of regions by a kind of limiting process, the precise notion of limit that we use today was not formulated until the 19th century by Cauchy and Weierstrass.
The main results of this chapter are the following:
The definition of the limit of a sequence,
The definition of the real number e (Theorem 2.3.),
The Squeeze Theorem (Theorem 2.5.),
the Bolzano Weierstrass Theorem (Theorem 2.8. and Theorem 2.10.),
The Cauchy Criterion (Theorem 2.9.),
the definition of an infinite series,
the Comparison Test (Theorem 2.17.), and
the Alternating Series Test (Theorem 2.18.).
These are powerful basic results about limits that will serve us well in later chapters.
A sequence of real or complex numbers is defined to be a function from the set N of natural numbers into the setR
or C.
Instead of referring to such a function as an assignment n→f(n), we ordinarily use the notation 
or 
Here, of course, an denotes the number f(n).
REMARK We expand this definition slightly on occasion
to make some of our notation more indicative.
That is, we sometimes index the terms of a sequence
beginning with an integer other than 1. For example, we write
or even 
We give next what is the most significant definition in the whole of mathematical analysis, i.e., what it means for a sequence to converge or to have a limit.
Let 
be a sequence of real numbers and let L be a real number.
The sequence 
is said to converge to L, or that L is the limit of 
, if the following condition
is satisfied.
For every positive number ϵ, there exists a natural number N such that
if n≥N,
then |an–L|<ϵ.
In symbols, we say L=liman or
We also may write an↦L.
If a sequence 
of real or complex numbers
converges to a number L,
we say that the sequence 
is convergent.
We say that a sequence 
of real numbers diverges to +∞ if for every
positive number M, there exists a natural number N such that
if n≥N,
then an≥M.
Note that we do not say that such a sequence
is convergent.
Similarly, we say that a sequence 
of real numbers diverges to –∞
if for every real number M, there
exists
a natural number N such that if
n≥N, then an≤M.
The definition of convergence for a sequence 
of complex numbers is
exactly the same as for a sequence of real numbers.
Thus, let 
be a sequence of complex numbers and let L be a complex number.
The sequence 
is said to converge to L, or that L is the limit of 
if the following condition
is satisfied.
For every positive number ϵ, there exists a natural number N such that
if n≥N,
then |zn–L|<ϵ.
REMARKS The natural number N of the preceding definition surely depends on the positive number ϵ. If ϵ' is a smaller positive number than ϵ, then the corresponding N' very likely will need to be larger than N. Sometimes we will indicate this dependence by writing N(ϵ) instead of simply N. It is always wise to remember that N depends on ϵ. On the other hand, the N or N(ϵ) in this definition is not unique. It should be clear that if a natural number N satisfies this definition, then any larger natural number M will also satisfy the definition. So, in fact, if there exists one natural number that works, then there exist infinitely many such natural numbers.
It is clear, too, from the definition that whether or not a sequence is convergent only depends on the “tail” of the sequence. Specifically, for any positive integer K, the numbers a1,a2,...,aK can take on any value whatsoever without affecting the convergence of the entire sequence. We are only concerned with an's for n≥N, and as soon as N is chosen to be greater than K, the first part of the sequence is irrelevant.
The definition of convergence is given as a fairly complicated sentence, and there are several other ways of saying the same thing. Here are two: For every ϵ>0, there exists a N such that, whenever n≥N,|an–L|<ϵ. And, given an ϵ>0, there exists a N such that |an–L|<ϵ for all n for which n≥N. It's a good idea to think about these two sentences and convince yourself that they really do “mean” the same thing as the one defining convergence.
It is clear from this definition that we can't check whether a sequence converges or not unless we know the limit value L. The whole thrust of this definition has to do with estimating the quantity |an–L|. We will see later that there are ways to tell in advance that a sequence converges without knowing the value of the limit.
Let an=1/n, and let us show that liman=0. Given an ϵ>0, let us choose a N such that 1/N<ϵ. (How do we know we can find such a N?) Now, if n≥N, then we have
which is exactly what we needed to show to conclude that 0=liman.
Let an=(2n+1)/(1–3n), and let L=–2/3. Let us show that L=liman. Indeed, if ϵ>0 is given, we must find a N, such that if n≥N then |an+(2/3)|<ϵ. Let us examine the quantity |an+2/3|. Maybe we can make some estimates on it, in such a way that it becomes clear how to find the natural number N.
for all n≥1. Therefore, if N is an integer for which N>1/ϵ, then
whenever n≥N, as desired. (How do we know that there exists a N which is larger than the number 1/ϵ?)
Let 
and let us show that
liman=0.
Given an ϵ>0, we must find an integer N
that satisfies the requirements of the definition.
It's a little trickier this time to choose this N.
Consider the positive number ϵ2.
We know, from Exercise 1.16, that there exists a
natural number N such that 1/N<ϵ2.
Now, if n≥N, then
which shows that 
REMARK A good way to attack a limit problem is to immediately examine the quantity |an–L|, which is what we did in Example 2.2 above. This is the quantity we eventually wish to show is less than ϵ when n≥N, and determining which N to use is always the hard part. Ordinarily, some algebraic manipulations can be performed on the expression |an–L| that can help us figure out exactly how to choose N. Just know that this process takes some getting used to, so practice!
Using the basic definition, prove that lim3/(2n+7)=0.
Using the basic definition, prove that lim1/n2=0.
Using the basic definition, prove that 
HINT: Use the idea from the remark above; i.e.,
examine the quantity |an–L|.
Again, using the basic definition, prove that
Remember the definition of the absolute value of a complex number.
Using the basic definition, prove that
Let an=(–1)n.
Prove that 1 is not the limit of the sequence 
HINT: Suppose the sequence 
does converge to 1.
Use ϵ=1, let N be the corresponding integer that exists in the definition,
satisfying |an–1|<1 for all n≥N,
and then examine the quantity |an–1| for various n's to get a contradiction.
Let 
be a sequence of (real or complex) numbers, and let L be a number.
Prove that L=liman if and only if
for every positive integer k there exists an integer