Modern Analysis, Homework 1
Due July 11, 2011
1. [20] (3 points) Rudin, Ch 7, #1: Prove that every uniformly convergent sequence of
bounded functions is uniformly bounded.
2. Rudin, Ch 7, #2: If
{
f
n
}
and
{
g
n
}
converge uniformly on a set
E
,
(a) [20] (2 points) Prove that
{
f
n
+
g
n
}
converges uniformly on
E
.
(b) [20] (3 points) If, in addition,
{
f
n
}
and
{
g
n
}
are sequences of bounded functions,
prove that
{
f
n
g
n
}
converges uniformly on
E
.
3. [25] (2 pts) Rudin, Ch 7, #3:
Construct sequences
{
f
n
}
and
{
g
n
}
which converge
uniformly on some set
E
but such that
{
f
n
g
n
}
does not converge uniformly on
E
(of
course,
{
f
n
g
n
}
must converge on
E
).
4. Rudin, Ch 7, #10 (mostly): Let
{
x
}
denote the fractional part of the real number
x
(by definition,
{
x
}
=
x
 b
x
c
, where
b
x
c
is the
floor function
–the greatest integer less
than or equal to
x
). For real
x
, define
f
(
x
) =
∞
X
n
=1
{
nx
}
n
2
.
(a) [31] (4 points) Find all discontinuities of
f
, and note that they form a countable
dense set.
Show that
f
is nevertheless Riemannintegrable on every bounded
interval
1
.
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 Summer '11
 Staff
 Metric space, 2 pts, Baire space, Baire Category Theorem, Nowhere dense set, Dense set

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