example of contractive sequence
Define the sequence by
for every : It’s clear when . If it is true for an , it implies that , i.e. .
As for the convergence of the sequence, which is not monotonic, one could think to show that it is a Cauchy sequence. Unfortunately, it is almost impossible to directly express and estimate the needed absolute value of . Fortunately, the recursive definition (1) allows quite easily to estimate . Then it turns out that it’s a question of a contractive sequence, whence it is by the parent entry (http://planetmath.org/ContractiveSequence) a Cauchy sequence.
We form the difference
where . Thus we can estimate its absolute value, by using (2):
Since , our sequence (1) is contractive, consequently Cauchy. Therefore it converges to a limit .
From the quadratic equation we get the positive root . I.e.,