How to prove that the following is Cauchy?

Question

So I have to prove that the following is a Cauchy sequence by definition:

$|u_{n+2}-u_{n+1}|\leq \frac{1}{2}|u_{n+1}-u_n|$.

I know that, $$|u_{n+2}-u_{n+1}|\leq \frac{1}{2^n} |u_2-u_1|$$

How do I proceed further?

Hint: $\vert u_m-u_n\vert\leq\sum_{k=n}^{m-1}\vert u_{k+1}-u_k\vert$. (Also, justify this inequality!) — Vercassivelaunos, Apr 24 '21 at 08:58

score 2 · Accepted Answer · answered Apr 24 '21 at 08:57

For $m \ge n \ge 2$, note that

\begin{align} |u_m - u_n| &= |u_m - u_{m - 1} + u_{m - 1} - \cdots + u_{n + 1} - u_n| \\ & \le |u_m - u_{m - 1}| + \cdots + |u_{n + 1} - u_n| \\ & \le \frac{1}{2^{m-2}}|u_2 - u_1| + \cdots + \frac{1}{2^{n-1}}|u_2 - u_1| \\ & = |u_2 - u_1|\left(\frac{1}{2^{n-1}} + \cdots + \frac{1}{2^{m - 2}}\right) \\ & \le |u_2 - u_1|\left(\frac{1}{2^{n-1}} + \cdots \right) \\ & = \frac{|u_2 - u_1|}{2^{n - 2}}. \end{align}

Can you conclude now?

How to prove that the following is Cauchy?

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