Define a sequence, $\displaystyle a_1=\frac{1}{2}$ and $\displaystyle a_{n+1}=\frac{1}{2+a_n}$

Prove that $\displaystyle |a_{n+1}-a_{n+2}|<\frac{|a_n-a_{n+1}|}{4}$

Proof by induction

When n=1, it is true

$\displaystyle \frac{1}{60}<0.04$

Assume true for n=k

$\displaystyle |a_{k+1}-a_{k+2}|<\frac{|a_k-a_{k+1}|}{4}$

Test n=k+1

$\displaystyle |a_{k+2}-a_{k+3}|<\frac{|a_{k+1}-a_{k+2}|}{4}$

$\displaystyle 4|\frac{1}{2+a_{k+1}}-\frac{1}{2+a_{k+2}}|<|\frac{1}{2+a_k}-\frac{1}{2+a_{k+1}}|$

$\displaystyle 4|\frac{a_{k+1}-a_{k+2}}{(2+a_{k+1})(2+a_{k+2})}|<|\frac{a_{k}-a_{k+1}}{(2+a_{k})(2+a_{k+1})}|$

By hypothesis, this is true for the numerators, now testing the denominators

if

$\displaystyle (2+a_{k+1})(2+a_{k+2})>(2+a_k)(2+a_{k+1})$ it is always true

Simplifying

$\displaystyle a_{k+2}>a_k$

However, this is not always true.

Now I need help. I am unsure how to proceed, or any of the preceding steps are wrong and I need to find a new method t solve the problem.

All help greatly appreciated