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.