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In this section we revise the notion of convergence for real numbers
and prove the BolzanoWeierstrass property.
From section 1 we have a least upper bound (greatest lower bound) for
any bounded increasing (respectively decreasing) sequence of real
numbers.
Exercise 28
Show that any bounded nondecreasing sequence of real numbers has a
least upper bound; a bounded nonincreasing sequence has a greatest
lower bound.
Now if S is any bounded nonempty set of real numbers, let c_{1} be an
upper bound and s_{1} S. Now we iteratively define a pair of
sequences {s_{n}} and {c_{n}} as follows. If
(s_{n} + c_{n})/2 is an
upper bound for S then we define
s_{n + 1} = s_{n} and
c_{n + 1} = (s_{n} + c_{n})/2; otherwise let s_{n + 1} be any element of S
so that
s_{n + 1} > (s_{n} + c_{n})/2 and
c_{n + 1} = c_{n}.
Exercise 29
Show that {
s_{n}} is a nondecreasing sequence of elements of
S
and {
c_{n}} is a nonincreasing sequence of upper bounds for
S
so that
(
c_{n + 1} 
s_{n + 1})
(
s_{n} 
c_{n})/2. Hence show that the
greatest lower bound of {
c_{n}} is equal to the least upper bound
of {
s_{n}} and this bound is the least upper bound for
S.
In particular, if {x_{n}} is any bounded sequence of real numbers we
have a least upper bound and greatest lower bound for this sequence.
Let us define
l_{k} 
= 
the greatest lower bound of {x_{n} n k} 

u_{k} 
= 
the least upper bound of {x_{n} n k} 

liminf{x_{n}} 
= 
the least upper bound of {l_{k}} 

limsup{x_{n}} 
= 
the greatest lower bound of {u_{k}} 

Note that {l_{k}} is a nondecreasing sequence and {u_{k}} is a
nonincreasing sequence.
Exercise 30
Show that
liminf{
x_{n}}
limsup{
x_{n}}.
We say that the sequence {x_{n}} has a limit (is convergent) if
these two numbers are equal; this number c is called the limit
of this sequence of numbers.
Exercise 31
Show that for every positive
there is a index
n_{0} so
that

x_{n} 
c <
for all
n >
n_{0}. Hence, there is an
index
n_{1} so that

x_{n} 
x_{m} <
for all
n >
n_{1}; this
called Cauchy's criterion. Conversely show that any
sequence {
x_{n}} that satisfies Cauchy's criterion is convergent.
Now for any sequence {x_{n}} we can find subsequences
{y_{k} = x_{nk}} (with
n_{1} < n_{2} < ^{ ... }) so that {y_{k}} is
convergent (this is called the BolzanoWeierstrass property).
Exercise 32
Show that there are subsequences of {x_{n}} that converge to
liminf{x_{n}} and
limsup{x_{n}}.
Finally, let us note some algebraic properties of convergent
sequences.
Exercise 33
Show that the sum, difference and product of convergent sequences is
limit of the sum, difference and product of the terms. If a sequence
has a nonzero limit then show that the inverse of the limit is the
limit of the inverses of the nonzero terms of the sequence.
Next: Functions, continuity and differentiability
Up: Prerequisites
Previous: Polynomials in more than
Kapil H. Paranjape
20010120