The definition of Cauchy sequence is: for any $ε > 0$, there exists a natural number $N$ such that if $m, n ≥ N$, then $|a_m − a_n| < ε$.
What if we changed the definition to: for any $k ≥ 1$, there exists a natural number $N$ such that $|a_{n+k} - a_n| < ε$ for any $n ≥ N$.
What is the difference between these 2 definitions. Will the second one be true as well?
Thank you!
$a_n=\ln n$ satisfies the second criterion but not the first. The first one implies the second by triangle inequality. [$|a_{n+k}-a_n|\leq |a_{n+k}-a_{n+k-1}|+|a_{n+k-1}-a_{n+k-2}|+\cdots |a_{n+1}-a_n|$. Make each term less than $\epsilon /k$].