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multimi marginite

 
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ciuprian



Joined: 26 Oct 2017
Posts: 1

PostPosted: Thu Oct 26, 2017 6:45 pm    Post subject: multimi marginite Reply with quote

O rugaminte!

cum putem demostra asa ceva?

o multtime A nevida este marginita daca si numai daca exista
un numar M > 0 astfel ıncat ∣x∣ ≤ M, ∀x ∈ A.

Multumesc!
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Laurian Filip
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Joined: 25 Nov 2007
Posts: 473
Location: Bucuresti

PostPosted: Thu Nov 02, 2017 2:10 pm    Post subject: Reply with quote

Ne amintim urmatoarele:

Definitia 1. O multime nevida A \subset \mathbb R se numeste minorata sau marginita inferior daca exista m \in \mathbb R \! astfel īncāt m \le a, \; \forall a \in A.
Definitia 2. O multime nevida A \subset \mathbb R se numeste majorata sau marginita superior daca exista M \in \mathbb R \! astfel īncāt M \le a, \; \forall a \in A.
Definitia 3. O multime nevida A \subset \mathbb R se numeste marginata daca este simultan marginata superior si inferior, adica daca exista m,M\in \mathbb{R} astfel incat m \le a \le M, \; \forall a \in A.


Revenind la intrebarea ta, rezolvam dubla implicatie separat:

\RightarrowStim ca A este marginata, deci exista m,M\in \mathbb{R} astfel incat m \le a \le M, \; \forall a \in A. Considerand acum c=max(|M|,|m|), este usor de vazut ca c>0 si  |x|\leq c, \forall x \in A.

\LeftarrowStiind ca exista M>0 astfel incat |x|\leq M, \forall x \in A, avem -M\leq x \leq M , \forall x \in A si deci multimea este marginata.
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