Problem solving seminar IV
14. Let fn : R → R be differentiable functions with |f (x)| ≤ 1 for all x ∈ R and
n ≥ 1. Assume
lim
n→∞
fn(x) = g(x)
for all x. Prove that g : R → R is continuous.
15. Show that the interval [0, 1] cannot be written as a countably infinite disjoint
union of closed subintervals of [0, 1].
16. Let N be a linear operator on an n-dimensional vector space, n > 1, such that
Nn
= 0, Nn−1
= 0. Prove that there is no operator X with X2
= N.
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