Let a0 and a1 be distinct real numbers.

Define an=((an-1)+(an-2))/2 for each positive integer n>=2.

Show that {an} is a Cauchy sequence...

Hint:

Use induction to show that

(an+1 - an)=(-1/2)^n*(a1-a0) and then use the result that 1+x+x^2+...+x^n=(1-x^(n+1)/(1-x)) if x=/=1.