Let $(x_n)_{n\in \mathbb{N}}$ be a sequence of real numbers such that the sequences \begin{align} y_n = x_{2n}, \quad z_n = x_{2n+1}, \quad w_n = x_{3n} \end{align} are Cauchy sequences. I need to prove that $(x_n)_n$ is also Cauchy.
How can I prove this?
My trial: Let $p\in \mathbb{N}$, $\varepsilon >0$ be arbitrary. I want to show that \begin{align} |x_{n+p} - x_n| < \varepsilon \end{align} for sufficiently large $n$. If $n$ is even, I considered the cases where $p$ is even and $p$ is odd. If $p$ is even, we are done as $(y_n)_n$ is Cauchy. But I could not proceed if $p$ is odd.
Is there any alternative approach?
Thanks in advance.