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2013/12/01

real number system 9 (bounded monotone sequences)

In the method of nested intervals, we supposed the following proposition.

If a real number sequence increasing or decreasing monotonically is bounded,  it will converge.

We shall prove it simply. A monotonical sequence is either

(increasing)\quad a_1\leq a_2\leq\cdots\leq a_n\leq\cdots , or
(decreasing)\quad a_1\geq a_2\geq\cdots\geq a_n \geq\cdots .

We will adderess the case of "a increasing sequence", as both cases are same.

If a increasing sequence is bounded above, it has \sup a_n=A (by the real number system property). Therefore for any \epsilon>0,
  A-\epsilon < a_m\leq a_{m+1}\leq \cdots \leq A  

Namely for all n such that n>m , |a_n-A|<\epsilon  because A-\epsilon<a_n\leq A .
As it means \lim_{n\rightarrow\infty}a_n=A , we get the desired result.



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