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Example text

H. a ~ b ist richtig, wenn entweder a < b oder a = b; • a:;;' b (lies: a gr6Ber oder gleich b): 23 1. 1 Grundlagen In diesem Sinne sind die Aussagen 4

L. 2, tiber Ordnungsrelationen in I, 1. 2 • 4 . 24 1. Elementare reelle Funktionen Satz Ftir beliebige a, b,c,d E IR ergibt die Addition zweier "Kleiner-als"-Ungleichungen wieder eine "Kleiner-als"-Ungleichung: aa+c a + c < b + C (additive Monotonie) cc+ba + c 14 -3 <-21\ -4 < 5, aber (-6)· (-4) > (-2)·5 Allerdings kann die Multiplikation unter eingeschriinkten Voraussetzungen erfolgen: Satz Sind a,d E IR; b,c E IR+, so gelten aa·c b 1\ c >d => a· c > b· d Beweis (fUr ,,<,,): Man bestiitigt sofort aO=>a·cO=>c·b a • c < b • d (Transitivitiit von II < II) Es empfiehlt sich zu merken, daB diese Multiplikation zweier gleichgerichteter Ungleichungen insbesondere dann gestattet ist, wenn aIle beteiligten GroBen positiv sind.

7,3>5,4 2. h. -14> -15 3. 13 > -4 => 13 - 19 > -4 - 19; d. h. -1>-3 4. ~,. h. a ~ b ist richtig, wenn entweder a < b oder a = b; • a:;;' b (lies: a gr6Ber oder gleich b): 23 1. 1 Grundlagen In diesem Sinne sind die Aussagen 4

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