Math 55a: Honors Advanced Calculus and Linear Algebra
Revised for Math 25b, Spring [2013–]2014
1
Metric topology IV: Sequences and convergence;
the spaces
B
(
X, Y
) and
C
(
X, Y
), and uniform convergence
Sequences and convergence in metric spaces.
[See Simmons, pages 70–71;
we’ll discuss Cauchy sequences etc. only next week, and won’t cover Theorems
C, D, E on pages 72–74 at all.] The notion of convergence in the metric space
R
is implicit in such familiar contexts as infinite series and even nonterminating
decimals, but was not made explicit until centuries after Euler spent considerable
effort trying to evaluate such series as 1!

2! + 3!

4! +
· · ·
[sic]. As with many
such notions, while our initial interest is in sequences in
R
or perhaps
R
n
, the
basics are just as easy to formulate in the context of an arbitrary metric space,
and we shall have occasion to use this notion in that generality when studying
sequences in function spaces (such as Taylor series).
Let
{
p
n
}
=
{
p
1
, p
2
, p
3
, . . .
}
2
be a sequence in a metric space
X
(i.e., with each
p
n
∈
X
,
n
= 1
,
2
,
3
, . . .
). We say that
{
p
n
}
converges
if there is a point
p
∈
X
such that: for every
>
0 there is an integer
N
such that
d
(
p
n
, p
)
<
for each
n > N
. Equivalently, for every
>
0 we have
d
(
p
n
, p
)
<
for all but finitely
many
n
. [Why are the two definitions equivalent?] Such
p
is called the
limit
of
{
p
n
}
. The definite article must be justified: we must show that if
p, p
0
are both
limits of
{
p
n
}
then
p
=
p
0
. This is easily shown as follows [Simmons suggests
another approach on pages 70–71]. For any
>
0, there are integers
N, N
0
such
that
d
(
p
n
, p
)
<
for each
n > N
and
d
(
p
n
, p
0
)
<
for each
n > N
0
.
Let
n
be any integer that exceeds both
N
and
N
0
.
Then by the triangle inequality
d
(
p, p
0
)
<
2 . Since
is an arbitrary positive number, it follows that
p
=
p
0
. We
also use the following notations for “
p
is the limit of
{
p
n
}
”: “
{
p
n
}
converges
to
p
”, “
p
n
approaches
p
”, “
p
= lim
n
→∞
p
n
”, and “
p
n
→
p
” (or “
p
n
→
p
as
n
→ ∞
”).
3
A sequence that does not converge is said to
diverge
. Note that this notion can
depend on
X
as well as
{
p
n
}
; e.g.
{
2

n
}
converges as a sequence in
R
, but not
as a sequence in (0
,
1). We can say “
{
p
n
}
converges in
X
” if the ambient space
may otherwise be ambiguous.