(1) In Section 1.4 we used the Axiom of Completeness (AoC) to prove the Archimedean Property of R (Theorem 1.4.2). Show that the
Monotone Convergence Theorem can also be used to prove the Archimedean Property without making any use of AoC.
(2) Use the Monotone Convergence Theorem to supply a proof for the Nested Interval Property (Theore 1.4.1) that doesn't make use of AoC. These two results suggest that we could have used the Monotone Con- vergence Theorem in place of AoC as our starting axiom for building a proper theory of the real numbers
(2) Use the Monotone Convergence Theorem to supply a proof for the Nested Interval Property (Theore 1.4.1) that doesn't make use of AoC. These two results suggest that we could have used the Monotone Con- vergence Theorem in place of AoC as our starting axiom for building a proper theory of the real numbers
1 answer:
You might be interested in
Given:
In circle O, m∠R = 30.8°.
To find:
The m∠NOQ
Solution:
Central angle theorem: According to this theorem, the central angle is always twice of subtended angle on the same arc.
Angle NOQ and angle NRQ are on the same arc but ∠NOQ is the central angle and ∠NRQ is the subtended angle on the arc NQ.
Using central angle theorem, we get
Therefore, .