517 Midterm 2
1. Define fn : [0, 1] → R by fn (x) = xn . Show {fn } converges pointwise but not uniformly to
(
0, x ∈ [0, 1)
f (x) =
1, x = 1.
Solution. For x ∈ [0, 1), limn→∞ xn = 0, and for x = 1 of course the limit isp1. This shows
pointwise convergence to f . Let = 1/2 and take any N . Then for x = N 1/2, we have
|fN (x) − f (x)| = 1/2 ≥ . This shows the convergence is not uniform.
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2. Let f : [0, ∞) → R such that f (n) (0) = 0 for n = 0, 1, 2, . . . and {f (n) } is uniformly bounded.
Show that f ≡ 0.
Solution. Let M > 0 be such that |f (n) (x)| ≤ M for all x ∈ [0, ∞) and all n ≥ 0. By Taylor’s
theorem, for any n ≥ 1,
f (n) (ζ )xn M xn
n
|f (x)| = .
≤
n!
n!
Letting n → ∞, we find that f (x) = 0.
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3. Let fn : X → R be uniformly continuous for each n, and suppose {fn } converges uniformly
to f . Show that f is uniformly continuous.
Solution. Let > 0. Pick N such that n ≥ N implies |fn (x) − f (x)| < /3 for every x ∈ X.
Pick δ > 0 such that |fN (x) − fN (y)| < /3 for all x, y ∈ X with d(x, y) < δ. Then for any
x, y ∈ X with d(x, y) < δ,
|f (x) − f (y)| ≤ |f (x) − fN (x)| + |fN (x) − fN (y)| + |fN (y) − f (y)| < .
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4. Let fn : [0, 1] → R be such that fn (0) = 0 and there is K > 0 such that |fn (x) − fn (y)| ≤
K|x − y| for all x, y ∈ [0, 1] and all n = 1, 2, . . .. Show that {fn } has a uniformly convergent
subsequence.
Solution. Note that {fn } is equicontinuous: given > 0, for δ = /K we have |fn (x) − fn (y)| ≤
K|x − y| < for all n and all x, y ∈ [0, 1] with |x − y| < δ. Also, {fn } is uniformly bounded:
fn (0) = 0 and the Lipschitz condition show that |fn (x)| ≤ K for all x ∈ [0, 1] and all n. Since
[0, 1] is compact, Arzela-Ascoli shows that {fn } has a uniformly convergent subsequence.
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5. Let fn : X → R be uniformly continuous for each n. Show that if {fn } converges uniformly,
then {fn } is equicontinuous.
Solution. Let > 0. Suppose {fn } converges uniformly to f , and pick N such that n, m ≥
N implies |fn (x) − fm (x)| < /3 for all x ∈ X. For i = 1, . . . , N , pick δi > 0 such that
|fi (x) − fi (y)| < /3 for all x, y ∈ X with d(x, y) < δ. Then for x, y ∈ X with d(x, y) < δ,
(
|fn (x) − fN (x)| + |fN (x) − fN (y)| + |fN (y) − fn (y)| < , n > N
|fn (x) − fn (y)| ≤
.
/3 < ,
n ∈ {1, . . . , N }
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