# Thread: Proof of contractive sequence

1. ## Proof of contractive sequence

A sequence (Sn) is said to be contractive if there exists k with 0<k<1 such that |S(n+2) - S(n+1)| <= k|S(n+1) - Sn| for all n is an element of N. Prove that every contractive sequence is a Cauchy sequence, and hence is convergent

2. Originally Posted by learn18
A sequence (Sn) is said to be contractive if there exists k with 0<k<1 such that |S(n+2) - S(n+1)| <= k|S(n+1) - Sn| for all n is an element of N. Prove that every contractive sequence is a Cauchy sequence, and hence is convergent
I did not see the problem. But do not accuse other people on this forum by not helping you . Yes, this is a harder problem. I am not going to write out the full proof, the general idea is contained in the image below.