PROBLEM SET V
“NOWHERE NEAR THE FINAL FRONTIER”
DUE FRIDAY, 21 OCTOBER
Exercise 36. Suppose s ∈ R. Show that the function rs : (0, ∞)
(0, ∞) given by rs (x) = xs is continuous.
Exercise 37. Show that if K ⊂ R is compact, then there exists a real number N > 0 such that K ⊂ [−N, N ].
Exercise 38. Suppose I1 ⊃ I2 ⊃ · · · a sequence of closed (bounded) intervals. Show that the intersection
T
∞
k=1 Ik 6= ∅.
Definition. The diameter of a bounded subset E ⊂ R is the real number
diam(E) := sup{|x − y| | x, y ∈ E}.
Exercise? 39. Suppose K ⊂ R a compact subset, and suppose {Uα }α∈Λ an open cover of K. Show that
there exists a real number λ such that for any subset V ⊂ K such that diam(V ) < λ, there exists an element
α ∈ Λ such that V ⊂ Uα .
Definition. Suppose E ⊂ R. Then a map f : E
R is uniformly continuous if for any ε > 0, there exists
a quantity δ > 0 such that for any points x0 , x1 ∈ E, if |x0 − x1 | < δ, then |f (x0 ) − f (x1 )| < ε.
Exercise 40. Suppose E ⊂ R. Show that a uniformly continuous map E
that the converse holds if E is compact.
1
R is continuous, and show