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## Homework Statement

Consider the sequence (x(n)) defined recursively by x(1)=1 and x(n+1)=1+1/x(n).

Prove that abs[x(n+2)-x(n+1)]<= 1/2 abs[x(n+1)-x(n)].

## Homework Equations

## The Attempt at a Solution

Proof by induction.

For n=1, abs(x(3)-x(2))<= 1/2 abs(x(2)-x(1))

abs(-1/2)<=1/2 abs(1)

1/2=1/2

So, for n=1 is true.

Suppose that n=k and that

abs[x(k+2)-x(k+1)]<= 1/2 abs[x(k+1)-x(k)]. We have to show that

abs[x((k+1)+2)-x((k+1)+1)]<= 1/2 abs[x((k+1)+1)-x(k+1)].

I'm not sure if this is correct. If so, how can I continue?

Thanks in advance!

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