 
Summary: Math 311001 201110
Assignment # 3 (due: March 10th)
1. Prove that every uniformly convergent sequence of bounded functions
is uniformly bounded.
2. If {fn}, {gn} converge uniformly on E, prove that so does {fn + gn}.
3. Construct sequences {fn}, {gn} such that both converge uniformly on
a set E, but such that {fngn} does not converge uniformly. Can you
make all functions fn, gn bounded?
4. Consider
f(x) =
n=1
1
1 + n2x
.
For what values of x does the series converge absolutely? On what
intervals does it converge uniformly? On what intervals does if fail to
converge uniformly? Is f continuous wherever the series converges? Is
f bounded?
5. Let
