t***y 发帖数: 110 | 1 有上界一定有上确界。递增不是必须的。
好象实数有八条重要的性质,以任何一个开头
作为原理都可以推出另外7条。
欧姆书上是用有界必有上确界作原理的。 和和。 | x*****d 发帖数: 427 | 2 修正一下,按小数定义时不这么简单。首先要定义实数的Cauchy sequnce,
然后证明每个实数cauchy sequnce收敛到一个实数,然后才有递增有上界
序列是cauchy sequence因而收敛到一个实数,这个实数就是上确界。 | t***y 发帖数: 110 | 3 I did not say it is a Gongli, I just said it is a Yuanli.
the basic properties of the reals are equivalent. We can
start from any one to prove others. Many analysis text
books also just pick one as a Yuanli (without using the
definition of the reals) to prove others.
Sorry for did not check the text book carefully last night.
hehe.
Prof Zhang's text books indeed using 规范小数 gives the
proof of Quejie Yuanli(第二种陈述): (P28-P31)
R 的任何一个非空并且有下界的子集合E在R中有下确界。
Also This theorem just states the property
【在 x*****d 的大作中提到】 : 修正一下,按小数定义时不这么简单。首先要定义实数的Cauchy sequnce, : 然后证明每个实数cauchy sequnce收敛到一个实数,然后才有递增有上界 : 序列是cauchy sequence因而收敛到一个实数,这个实数就是上确界。
|
|