TrueFalse prove or find a counterexample If xn is a Cauchy s

True/False prove or find, a counterexample: If {x_n} is a Cauchy sequence then there exists an M such that for all n greaterthanorequalto M we have |x_n + 1 - x_n| lessthanorequalto |x_n - x_n - 1|.

Solution

Let AR be a nonempty set, bounded from above by x. We will show that A has a least upper bound. Let a0A and b0:=x. Inductively define sequences (an), (bn) as follows: If an+bn2 is an upper bound of A, let an+1:=an, bn+1:=12(an+bn). Otherwise, choose an+1A with 12(an+bn)an+1bn, and let bn+1:=bn Then: (an) is increasing, (bn) is decreasing, both are bounded, anA for all n and Abn for all n. Moreover (bnan)2n(b0a0). Now, by the monotonicity, an converges, say ans. As (bnan)2n(b0a0)0, bns also. We will prove that s is the least upper bound of A. On one hand, if aA, then abn for all n, which implies alimnbn=s. Hence s is an upper bound. If t is any upper bound, as anA, we have ant for all n, giving s=limnant. Hence s is the least upper bound.