Answers & Hints for Selected Exercises 697- (a) \:In E N, [[fn[[ = +oo, so convergence is not uniform on R But on any
compact [a, b], the convergence is uniform since [[fn[[ =max { l~I, l~I}----> 0.
(b) \:In EN, llfn - J[[ = sup{[fn(x)-~[: x E JR}=~' so convergence is not
uniform on R But Va> 0, convergence is uniform on [a, oo) since [[fn - J[[ =
sup{[ tan-^1 nx - ~I: x;::: a}= tan-^1 na - ~----> 0. Similarly on (-oo, -a].(c) fn(x) ----> 0 on [O, oo). From f'(x) = n(~:;ix), we find that fn has its
max when x = 1/n. Thus, [[fn/l = fn(l/n) = 1 /e f+ 0, so convergence is
not uniform on [O, oo). But Va > 0, convergence is uniform on [a, oo) since
[[fn[[ = fn(a) = ;~ ----> 0 by L'Hopital's rule.(d) Convergence is uniform on any [a, b], where 0 <a< b < 2 or -2 <a<
b < 0, but is not uniform on any interval containing -2, 0, or 2, or on any
unbounded interval.(e) Convergence is uniform on any [a, b], where <^2 k;l)7r <a< b <^2 <ktl)7r,
but is not uniform on any interval containing any^2 <ktl)7r, k E Z.(f) Convergence is uniform on any [a, b], where (k - l)7r <a< b < k1f, but
is not uniform on any interval containing br, k E Z.(g) Convergence is not uniform on any interval containing more than one
point.(h) Convergence is uniform on any ( -oo, a] or [b, +oo), where a < 0 < b,
but is not uniform on any interval containing 0.(i) Convergence is uniform on any interval not containing -1 or 1.
(j), (1) Convergence is uniform on any compact [a, b].(k) Convergence is uniform on any [a, b], where 0 <a< b.
- [[Jn[[ = [[Jn - f + J[[ :S [[Jn - f[[ + [[f[[, SO [[Jn[[ - [[J[[ :S [[Jn - J[[. Also,
[[fll = [[f - fn + fn[[ :S [[f - fn[[ +[[Jn[[, SO [[f[[ - [[fn[[ :S llJ - fnll· Therefore,
I llfnll - llf[[ [ :S [[fn - f[[. So, fn ----> f uniformly==> llfn - fll ----> 0 ==> llfnll ---->
llfll·
oo n - (==>)Suppose I:; fk = f uniformly on S , and let Sn= I:; fk· Then Sn----> f
k=O k=O
uniformly on S. Let c > 0. Then :3no EN 3 n 2 no ==>Sn - f is bounded and
llSn - fll < c/2. Then m , n 2: no==> llSm - Snll :S l!Sn - fll +!If-Smll < c.
That is, m,n;::: no==> II f: jkll < c.
k=m+l