1.1. Supremum and Infimum of Sets of Real
Numbers. Continued Fractions
1.1.1. Show that
sup{x e Q : x 0 , x2 2} = V2.
1.1.2. Let A C M be a nonempty set. Define —A = {x : —x e A}.
Show that
sup(—A) = —inf A,
inf (—A) = - s u p A.
1.1.3. Let A, B e l be nonempty. Define
A + B = {z = x + y:xeA, y e B } ,
A-B = {z = x-y:xeA, y B}.
Show that
sup(A -f B) = sup A + sup B,
sup(A - B) = sup A - inf B.
Establish analogous formulas for inf (A + B) and inf (A B).
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