696 Appendix C • Answers & Hints for Selected Exercises
Chapter 9
EXERCISE SET 9.1- (a) and (c), are subspaces; the others are not.
- (a) fn(x)--+ 0 on R (c) fn(x)--+ 0 on [O,oo).
(b) lim fn(x) = -n/2 if x < 0, 0 if x = 0, n/2 if x > 0.
n-><XJ
(d) fn(x)--+OifxE(-2,0)u(0,2), lifx=O.
(e) fn(x) --+ 0 if x-=/= (^2 ktl)7r, 1 if x = (^2 ktl)7r, k E Z.
(f) fn converges pointwise on IR - {(2k + l)n: k E Z}: to 1 if x = 2kn,
to 0 if x-=/= kn, k E Z.
(g) fn(x) converges pointwise on IR to 0 if x = 0, 1 if x-=/= 0.
(h) fn(x)--+ 0 on R
(i) Converges pointwise on IR - {-1} to 0 if !xi< 1, 1 if !xi> 1,
1/2 if x = 1.
(j) fn(x)--+ x/3 on R (1) fn(x)--+ ex on R
(k) Converges pointwise on [O, oo) to 0 if x > 0, -1 if x = 0. - Yes. Each function fn(x) =Ix! if lxl 2: l/n, ~ - !x i if !xi < l/n, has a local
max at 0, but the limit function, f(x) = !xi does not.
7. Vn EN, f 0
1
fn = 3 + l/n, so lim f 0
1
fn = 3. But, f 0
1
lim fn = f 01
n-+<X> n-+cx:>^0 = 0.- (a) VO < x :::; 1, y'x --+ 1 by Thm. 2.3.9 .. ·.^2 n-..yx --+ 1, since it is a
subsequence.
(b) V - 1 :=:; x < 0, x l /(^2 n- l) = -(-x)^1 1(^2 n-l)--+ -1 by (a).
(c) x2n-l ~ = x l+-1-2n-1 = x. x2n-l 1 --+ { x if.^0 < x < -^1 ' } on [-1, 1].
- x if -1:::; x < 0
EXERCISE SET 9.2- (a) sup{!f(x)! : x ES} 2: 0, and= 0 ~ Vx ES, f(x) = 0.
(b) sup{!f(x) + g(x)I : x ES}:::; sup{!f(x)I : x ES}+ sup{!g(x)!: x ES}.
(c) sup{!rf(x)I: x ES}= !r!sup{lf(x)!: x ES}. - They are all equivalent.
- (b) Vn EN, llfn - f!I = 2n f+ 0 (d) Vn EN, llfn - fll = 1 f+ 0.
- Vx E IR, lsin (x + ~) - sin xi = I cosc!/n:::; l/n for some x < c < x + l/n.
:. llfn - f 11 :::; l /n--+ 0. - (a) and (d) converge uniformly on [O, +oo).