B U Department of Mathematics
Fall 2005 Math 331  Real Analysis I,
Second Midterm, 21/12/2005
1. (1) What does uniform convergence of a series of functions
∑
∞
n
=1
f
n
(
x
)
mean?
The uniform convergence of the sequence of its partial sums.
2. (1) Why do people sometimes require that
∑
∞
n
=1
f
n
(
x
)
is to be uniformly convergent?
It gives an answer to the question whether the limit function
f
(
x
) =
∑
∞
n
=1
f
n
(
x
)
is continuous.
3. (1) Can a uniformly continuous function on
R
be uniformly convergent?
Does not make sense.
4. (1) Which country was Abel from?
Norway.
5. (8) Suppose
(
M, d
)
and
(
N, u
)
are two metric spaces and
f
:
M
→
N
is uniformly continuous.
Prove that a Cauchy sequence in
M
has its image under
f
a Cauchy sequence in
N
.
Solution:
Let
{
a
n
} ⊂
M
be Cauchy. We have:
•
Given
>
0
, there exists
δ >
0
such that whenever
d
(
x, y
)
< δ
,
u
(
f
(
x
)
, f
(
y
))
<
.
•
Given
δ >
0
, there exists
N
such that whenever
n, m
≥
N
,
d
(
a
n
, a
m
)
< δ
.
Then, given
, there exists
N
such that whenever
n, m
≥
N
we have
d
(
a
n
, a
m
)
< δ
so that
u
(
f
(
a
n
)
, f
(
a
m
))
<
.
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 Spring '11
 talinbudak
 Math, Calculus, Metric space, Compact space, Uniform convergence, Weierstrass Mtest

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