Problem : Let {$\\x_n\\$} ∞ n=1 and {$\\y_n\\$} ∞ n=1 be two sequences such that {$\\y_n\\$} ∞ n=1 converges to zero. Suppose that for all positive integers k and m with m ≥ k, we have |$\\x_m\\$ − $\\x_k\\$| ≤ $\\y_k\\$.
Prove that {$\\x_n\\$} ∞ n=1 is a Cauchy sequence.
I'm stuck from the start because the definition of cauchy sequence would require another xn - xk and I was wondering if using the definition of a Cauchy sequence, can I just assume m as the index n since n is not an integer? - which would allow me to apply the definition directly.
Not sure if im explaining my idea properly but I'm currently quite confused and not sure how to do this.
Any help would be appreciated thanks~
$m,n$ and $k$ are all positive integers. Let $ \epsilon >0$. Then there is $K \in \mathbb N$ such that $y_k < \epsilon$ for all $k>K$. Hence, if $k>K$ and $m \ge k$, we have
$$|x_m-x_k| \le y_k < \epsilon. $$
This shows that $(x_n)$ is a Cauchy sequence.