Functional Analysis
Let f be a nonnegative bounded function on [a, b] with
. Let
for n = 1, 2, … and 
and set
. Prove that
and that
for each n. Thus
converges uniformly to f on [a, b].
(Note that in this Exercise 1 there is
. Given a subset A of some “universal” set S, we define 
: S ® R, the characteristic function of A, by (x) = 1 if x is in A, and (x) = 0 if x is not in A.)
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