Given two sequences (perhaps are recursive functions instead) $a_n$ and $b_n$, defined on $N = [0,+\infty[$ which have the same initial term :
$a_0 = 0$
$b_0 = 0$
And are defined as follows :
$a_n = \frac {k - a_{n-1}} {c} + a_{n-1}$
$b_n = \frac {k - b_{n-1}} {d} + b_{n-1}$
with $k > 0$
Both sequences are increasing and converge to the value of $k$.
Given that $c < d$ how do I prove that $a_n \geq b_n$? or at least prove that the statement is correct for a given subset of $N$?
Knowing that $\frac {k - a_0} {c} > \frac {k - b_0} {d} $ from $c < d$, I could prove that $a_1 > b_1$. yet at one point, the condition $\frac {k - a_n} {c} > \frac {k - b_n} {d} $ will no longer hold since $a_n$ is converging faster to $k$ therefore at one point, the following will be true :
$\frac {k - a_n} {c} < \frac {k - b_n} {d} $
Trying it with concrete values always seems to verify that $a_n \geq b_n$ (in figures).
Any ideas on how to approach this problem? or atleast on how to define the ranges in which $a_n \geq b_n$ is applicable?
Thank you in advance !
for $a_n, k=10, c=2$
$k=10, c=2$" />
for $b_n, k=10, d=7$
$k=10, d=2$" />
generated through http://www.calcul.com/show/calculator/recursive