su p (1)=
ex
and, if inf A = 0, then sup (^-) = +oo. Additionally, show that if A
and B are bounded sets of real numbers, then
sup(A B)
= max{sup A sup B, sup A inf B, inf A sup B, inf A inf B}.
1.1.5. Let A and B be nonempty subsets of real numbers. Show
that
sup(A U B) = max{sup A, sup B}
and
inf (A U B) = min{inf A, inf B}.
1.1.6. Find the least upper bound and the greatest lower bound of
Ai, A2 defined by setting
Ai =
J2(-l)"+1
+ ( - 1 ) ^ (2 + £) : n N} ,
fn-1 2717T 1
A
2
= 7 cos —— : n e N .
\ n + l
3 J
1.1.7. Find the supremum and the infimum of the sets A and B,
where A = {0.2,0.22,0.222,... } and B is the set of decimal frac-
tions between 0 and 1 whose only digits are zeros and ones.
1.1.8. Find the greatest lower and the least upper bounds of the set
of numbers
\n^'
, where n e N.
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