Prove that if $|a_n|<2 \,\forall n\in \mathbb{N}$ and $|a_{n+2}-a_{n+1}|\leq \frac{1}{8}|a_{n+1}^2 - a_n^2|$, then $\{a_n\}$ converges.

Question

The problem is the following:

Prove that if $|a_n|<2 \,\forall n\in \mathbb{N}$ and $|a_{n+2}-a_{n+1}|\leq \frac{1}{8}|a_{n+1}^2 - a_n^2|$, then $\{a_n\}$ converges.

My attempt: Since $\{a_n\}$ is a sequence of reals, I think the relationships suggest proving that $\{a_n\}$ is a Cauchy sequence.

The two hypothesis imply this result:

$$|a_{n+2}-a_{n+1}|\leq \frac{1}{8}|a_{n+1}^2 - a_n^2| \leq \frac{1}{8}|a_{n+1}-a_n||a_{n+1}+a_n|\leq\frac{1}{8}|a_{n+1}-a_n|(|a_{n+1}| + |a_n|)<\frac{1}{8}|a_{n+1}-a_n|(2+2)=\frac{1}{2}|a_{n+1}-a_n|.$$

Using induction, is easy to see that

$$|a_{n+k}-a_{n+k-1}| \leq \frac{1}{2^{k-1}}|a_{n+1}-a_n|.$$

Now, let $n,m \in \mathbb{N}$, and suppose $n<m$, then $m = n +k$, for some $k \in \mathbb{N}.$ Now,

$$|a_m-a_n|=|(a_{n+k}-a_{n+k-1})+ (a_{n+k-1}-a_{n+k-2})+ \ldots+(a_{n+2}-a_{n+1})+(a_{n+1}-a_n)|,$$ so $$|a_m-a_n|\leq |a_{n+k}-a_{n+k-1}|+ |a_{n+k-1}-a_{n+k-2}|+ \ldots+|a_{n+2}-a_{n+1}|+|a_{n+1}-a_n|,$$ and then it follows that

$$|a_m - a_n| \leq |a_{n+1}-a_n|\big(\frac{1}{2^{k-1}}+\frac{1}{2^{k-2}}+ \ldots + \frac{1}{2^{2}}+\frac{1}{2}+1\big),$$

And I'm stuck here. My questions are:

1. Is all this process correct?
2. How can I conclude? Given $\varepsilon >0$, how do I choose $N$?

Answer 1

Looking at the way you have set up the problem, I would use the following equivalent condition for a sequence $(a_n)$ to be Cauchy:

$$\lim_{n \to \infty}|a_{n+k}-a_n| = 0, \quad \forall k \in \mathbb N. $$

You have already obtained

$$|a_{n+k}-a_n| \le C(k) |a_{n+1}-a_n|, $$

so you only need to show that $|a_{n+1}-a_n| \to 0$. But this shouldn't be hard for you, given what you have already done.

Answer 2

It is also possible, after the first line of the solution proposed by OP, to take series view of a sequence i.e. $a_n=\sum_0^{n-1} (a_{k+1}-a_k)$, and deduce convergence by proving absolute convergence of the series using comparison with geometric series.