The Sequence ${a_n}$ satisfies the following condition ::
$|a_{n+1} - a_n|$ $\leq$ $\alpha$$|a_n-a_{n-1}|$, $0<\alpha <1 $ for all natural numbers $n\geq2$
I approached the question like this ::
In a field of real numbers, A sequence is convergent iff it's cauchy. So I tried to prove that the sequence is convergent using monotone convergence theorem. Using Induction, I first proved that the sequence is monotonous. And again using induction, trying to prove that it is bounded, but I am stuck here.
Apart form this, I think applying Cauchy's general principle of convergence to prove this would be a more neater technique. Cauchy's General principle of Convergence states that the necessary and sufficient condition for convergence of a sequence ${S_n}$ is that for each $\epsilon$, there exists an $m$ such that
$|S_{n+p}-S_n| < \epsilon$, $\forall n>m$ and $p\geq1$
Please let me know how this can be approached by either of the ways. Thanks.
The argument is telescoping in nature as evident by what is given. See below a sketch. Also, recall that $\alpha^n \to 0$ when $0<\alpha < 1$
Suppose $m > n $, then
$$ |a_m - a_n| = | a_m - a_{m-1} + a_{m-1} - a_{m-2} + a_{m-2} - ... + a_{n+1} - a_n| \leq |a_m - a_{m-1}| + |a_{m-1} - a_{m-2}| + ... + |a_{n+1} -a_n| \leq \alpha^m |a_3-a_2| + \alpha^{m-1} |a_3-a_2| + ... + \alpha^n |a_3 - a_2| = \sum_{k=n}^m \alpha^k |a_3-a_2| $$