Let $a,b$ be positive real number. Set $x_0 = a$ and $x_{n+1} = 1/[(1/x_n) + b]$ for $n ≥ 0$
(a) Prove that $x_n$ is monotone decreasing.
(b) Prove that the limit exists and find it.
My work:
(a) By given premises, $x_n ≥ 0$ for each n,
$x_1 = x_{0+1} = 1/[(1/x_0) + b] = 1/[(1/a) + b]$
$x_2 = x_{1+1} = 1/[(1/x_1) + b] = 1/[(1/a) + b + b] = 1/[(1/a) + 2b]$
and then continues.
we can see as $n$ increases, $x$ dereases as denominator increases.
$n→∞, nb→∞$, the sequences decreases
(then i dont know how to continue with induction)
(b) By Monotone Convergence Theorem, the sequence ($x_n$) is convergent, bounded below, limit exists
....But I dont know how to find that limit ....
Thank you guys!!Please