最小上界定理證明

[rebe212296 (綠豆冰)]

The least-upper-bound property states that

any non-empty set of real numbers that has an upper bound

must have a least upper bound.


下面的證明是用柯西數列 可是我看不出哪邊有用到柯西數列 請版大指點 謝謝


Proof using Cauchy sequences


It is possible to prove the least-upper-bound property

using the assumption that every Cauchy sequence of real numbers converges.


Let S be a nonempty set of real numbers,

and suppose that S has an upper bound B1.

Since S is nonempty,

there exists a real number A1 that is not an upper bound for S.


Define sequences A1, A2, A3, ... and B1, B2, B3, ... recursively as follows:

Check whether (An + Bn) / 2 is an upper bound for S. (到這一行開始看不懂)

If it is, let An+1 = An and let Bn+1 = (An + Bn) / 2.

Otherwise there must be an element s in S so that s>(An + Bn) / 2.


Let An+1 = s and let Bn+1 = Bn.

Then A1 ≦ A2 ≦ A3 ≦ ...≦ B3 ≦ B2 ≦ B1 and |An - Bn| → 0 as n → ∞.


It follows that both sequences are Cauchy and have the same limit L,

which must be the least upper bound for S.

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推 suhorng:An跟Bn都是 Cauchy sequence
→ suhorng:(An+Bn)/2 那行在做二分搜
→ suhorng:你注意他的選法 每次都 An < Bn
→ suhorng:而且 An,Bn 越來越靠近
→ suhorng:而且 An, Bn 都把(想像中的)sup S夾在中間
推 gj942l41l4:畫個圖會比較清楚