How to prove that the following is Cauchy? 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?

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?
