Cauchy criterium

Question

Let $q\in[0,1),$ and let $a_n$ be a sequence with $|a_{n+1}−a_n |\leq q | a_n − a_{n−1} |$ for every $n\geq 2$ .

Show that $a_n$ is a Cauchy sequence.

I came to the conclusion that it sufficies to find $n_0$ such that:

$$[1/(1-q)]-q^n|a_2-a_1|<\varepsilon$$ for all $n\geq n_o$

How can I find $n_0$?

Answer 1

You can't find such a $n_0$, since $\lim_{n\to\infty}\dfrac1{1-q}-q^n\lvert a_2-a_1\rvert=\dfrac1{1-q}\neq0$.

Answer 2

Your conclusion is way off. For the proof you need two things:

(i)$\qquad|a_{n+1}-a_n|\leq q^n|a_1-a_0|\qquad(n\geq0)\ $;

(ii)$\qquad|a_{n+p}-a_n|\leq\sum_{k=n}^{n+p-1}|a_{k+1}-a_k|\ .$