548 Chapter 9 • Sequences and Series of Functions
sinnx- \:/n EN and \:/x E JR, define fn(x) = fa. Prove that
(a) {Jn} converges to the 0 function on (-oo,+oo).
(b) every f n is differentiable on ( -oo, +oo), and the limit function f has
derivative 0 everywhere on (-oo, +oo ).
(c) {!~}does not converge pointwise to f' on (-oo,+oo).- Prove that hm.^2 n-..yx = {^1. if^0 < x < -^1 ' }. (See Example 2.3.9.)
n-+oo -1 1f - 1 '.S X < 0
Use this to prove that the sequence { x^2 ~'.'..,} converges pointwise to [xi
on [-1, l].
9.2 Uniform Convergence
The notion of "pointwise" convergence ignores one of the essential features of
our definition of limit of a sequence. In Chapter 2, we agreed^3 that the statement
" lim Xn = x" means that Xn can be made arbitrarily "close to" x by making
n-+oo
n sufficiently large. We would like our definition of the limit statement
lim fn = f
n-+oo
to incorporate this notion of "closeness." We want it to mean that we can make
the function f n be arbitrarily "close to" the function f by making n sufficiently
large. But what does it mean to say that two functions are "close to" each
other? How do we measure the distance between two functions?
In defining closeness of real numbers we used the absolute value; thus,
[x -y[ represents the distance between x and y, and [xi represents the distance
between x and 0. To represent the "distance" between functions f and g we
can take a similar approach, as in the following definition.
Definition 9.2.1 Given a function f E B(S), we define
llfll = sup{[f(x)[ : x ES}.
The real number llfll is called the sup norm off on S; it is guaranteed to
exist by the completeness property. (See Figure 9.5.)
Given functions f , g E B(S), the distance between f and g on Sis
d(f, g) =Iii -g[[.- See Definition 2.1.4 and the verbal paraphrase that follows it.