First of all, a<, Cauchy sequence must be bounded, let a
Divide [a, b] into three parts, c=(2a+b)/3, d=(a+2b)/3. It is proved that there is an interval in [a, c] and [d, b] that contains at most finite items in the sequence.
If both intervals contain infinite terms in the sequence, then for e = (b-a)/3 >; 0 has n, when m>n> is n, there is | am-an | < E in [a].
There must be an ak, k >； in c]; n .
There must be an al, l > in [d, b]. N, then | AK-Al | > =(b-a)/3. Contradictions, so one of the two intervals contains a limited number of items at most.
Delete one with a limited number of terms (if both terms are limited, delete the left interval), and record the remaining intervals as [c 1, db 1]. Then [c 1, d 1] is divided into three parts, and one part is also removed, and then a closed interval sequence is obtained.
1, [cn, dn] contains [c(n+ 1), c(n+ 1)], and the interval length is (b-a)/3 n.
2. The exterior of [cn, dn] contains a limited number of items in the sequence {an}.
According to the theorem, cn and dn have a limit value x, which is in all closed intervals. Let's prove that x is the limit of {an}.
For any e>0, there is k, which makes CK
Pay attention to the second property, there are finite items with {an} outside [cK, dK], and remember that the maximum index is n, that is, n >；; When n and an are located in [cK,
DK], so | an-x |
Extended data
Cauchy convergence criterion properties of functions
1. sufficiency: since the function limit and the sequence limit can be linked by the resolution principle, proving the convergence of function can be transformed into proving the convergence of sequence. Cauchy criterion of sequence convergence has been proved, so it is the key to prove that the known conditions are transformed into finding the limit of sequence.
2. Resolution principle (or Heine theorem): Let f(x) be defined in the centripetal neighborhood of x0 (or when |x| is greater than a positive number), then the necessary and sufficient condition is that any sequence {xn} which converges to x0 and satisfies xn≠x0 or any sequence whose absolute value is greater than a positive number and diverges to infinity {
Source of reference: Baidu Encyclopedia-Cauchy Limit Existence Criterion
Reference source: Baidu Encyclopedia-Interval Set Theorem