such that for all
n
≥
N,

s
n

s

<
.
This is by definition equivalent
to
s
n
∈
N
(
s
)
.
Since by hypotheses
s
n
∈
F
for all
n,
and moreover we have assumed that
s
is not in
F,
consequently
s
n
∈
N
*
(
s
)
∩
F.
We have just shown that
s
satisfies the definition of being an accumulation
point of
F.
(why?) Hence,
s
∈
F,
by Theorem 13.17.
MATH 117 LECTURE NOTES FEBRUARY 17, 2009
5
(5) This is Theorem 16.13 in the text. Assume
{
s
n
}
∞
n
=1
is a convergent sequence
of real numbers and its limit is
s.
Then, by definition of convergence (geez,
go on and memorize it already!) using
= 1
,
there exists
N
∈
N
such that
for all
n
≥
N,

s
n

s

<
1
⇒ 
s
n

<

s

+ 1
for all
n
≥
N.
So define
M
=

s
1

+

s
2

+
. . .
+

s
N

1

+

s

+ 1
.
For any
n
≥
N,

s
n

<

s

+ 1
< M.
For any
n
≤
N

1
,

s
n

< M.
So the
sequence is bounded by
M.
6
DR. JULIE ROWLETT
(6) This was done in class on Tuesday February 10, 2009. The set of numbers
{
s
n
}
∞
n
=1
is bounded above by
s
1
and below by 0
.
Let
s
be the infimum of
S
=
{
s
n
}
∞
n
=1
.
Then, let
>
0
.
By definition of infimum (did you memorize
this?),
s
+
is no longer a lower bound for
S.
Hence, there is some
s
N
∈
S
such that
s
N
< s
+
.
However, the sequence is decreasing, so for all
n
≥
N,
s
n
≤
s
N
< s
+
.
By definition of infimum, for all
n
∈
N
,
s
≤
s
n
.
Hence, for all
n
≥
N,
s
≤
s
n
≤
s
+
⇒ 
s
n

s

<
.
This is the definition of convergence.
MATH 117 LECTURE NOTES FEBRUARY 17, 2009
7
2.
Monotone Sequences
Definition 2.1.
A sequence
{
s
n
}
∞
n
=1
is
increasing
if
s
n
≤
s
n
+1
for all
n
∈
N
.
The sequence is
decreasing
if
s
n
≥
s
n
+1
for all
n
∈
N
.
A sequence which is either increasing or decreasing (by the above definitions) is
called
monotone.
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 Fall '08
 Akhmedov,A
 Math, Order theory, Sn, Dominated convergence theorem, Monotone convergence theorem, DR. JULIE ROWLETT