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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