CM221A
ANALYSIS I
NOTES ON WEEK 2
You must remember and be able to use all the definitions and theorems stated in
this week’s notes.
Their proofs can be found in the CM115 lecture notes.
The
proofs are not examinable.
CONVERGENT SEQUENCES
Let
{
a
n
}
be a sequence.
The Crucial Definition
. We say that
{
a
n
}
converges to a number
a
and write
lim
n
→∞
a
n
=
a
or
a
n
→
a
if
∀
ε >
0
∃
n
ε
∈
N
:
n
≥
n
ε
⇒ 
a
n

a

< ε.
In usual words, this means the following: for every positive
ε
there is a positive
integer
n
ε
such that

a
n

a

< ε
whenever
n
≥
n
ε
. The number
a
is called the
limit
of
{
a
n
}
. A sequence which has a limit is called a
convergent
sequence.
One can reformulate the above definition in “geometric” terms.
The Crucial Definition: another version
. The sequence
{
a
n
}
converges to a
number
a
if for every
ε >
0 there are only finitely many elements
a
n
lying outside
the open interval (
a

ε, a
+
ε
).
Remark.
There are divergent sequences, that is, the sequences which do not
converge to a limit. For instance, the sequence
{
+1
,

1
,
+1
,

1
,
+1
, . . .
}
does not
converge.
Definition.
We say that
{
a
n
}
converges to +
∞
and write
a
n
→
+
∞
if
∀
R >
0
∃
n
R
∈
N
:
n
≥
n
R
⇒
a
n
> R.
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 Spring '09
 Mathematical analysis, 1 K, Limit of a sequence, Cauchy, 2 2m

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