Problem solving seminar V
17. Let f : [0, 1] → R be continuously differentiable with f(0) = 0. Prove that
sup
0≤x≤1
|f(x)| ≤
1
0
(f (x))2dx.
18. Prove or supply a counterexample: If f is a nondecreasing real valued function on
[0, 1], then there is a sequence {fn} of continuous functions on [0, 1] such that for each
x ∈ [0, 1]
lim
n→∞
fn(x) = f(x).
19. Let G be a group of order 10 which has a normal subgroup of order 2. Prove that
G is abelian.
20. Let G be a group and H and K subgroups such that H has a finite index in G.
Prove that K ∩ H has a finite index in K.
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