Problem solving seminar III
10. Suppose that fn is a sequence of nondecreasing functions which map the unit
interval into itself. Suppose that
lim
n→∞
fn(x) = f(x)
pointwise and that f is continuous function. Prove that fn(x) → f(x) uniformly as
n → ∞, 0 ≤ x ≤ 1. Note that the functions fn are not necessarily continuous.
11. Let G be a group and H a subgroup of index n < ∞. Prove or disprove the
following statements:
(A) If a ∈ G, then an
∈ H.
(B) If a ∈ G, then there is k, 1 ≤ k ≤ n such that ak
∈ H.
12. Let A be an n × n matrix and At
its transpose. Show that At
A and At
have the
same rank.
13. Let X ⊂ Rn
be compact and let f : X → R be continuous. Given ε > 0, show
that there is M such that for all x, y ∈ X
|f(x) − f(y)| ≤ M||x − y|| + ε.
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