TEMA 3 ROLES

Question 1

Then we have
∞
n=0 ∞
m=0
f(n, m)

= 
(n,m)∈N×N
f(n, m)
= 
(m,n)∈N×N
f(n, m)
= ∞
m=0 ∞
n=0
f(n, m)

.

Answer 1

There are of course infinitely many subsets of the real line; indeed,
Cantor’s theorem (Theorem 8.3.1; see also Exercise 8.3.4) shows that
there are even more such sets than there are real numbers. However,
there are certain special subsets of the real line (and the extended real
line) which arise quite often. One such family of sets are the intervals.
Definition 9.1.1 (Intervals). Let a, b ∈ R∗ be extended real numbers.
We define the closed interval [a, b] by
[a, b] := {x ∈ R∗ : a ≤ x ≤ b},
the half-open intervals [a, b) and (a, b] by
[a, b) := {x ∈ R∗ : a ≤ x<b></b>