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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