# What is the difference between this alternate definition of Cauchy and its actual definition? Will this one work?

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!

• You seem to have lost "for any $ε > 0$" in your second definition but then use $ε$. Condition order matters: "for any $ε > 0$ and for any $k ≥ 1$, there exists a natural number $N$ such that ..." is a weaker condition than "for any $ε > 0$, there exists a natural number $N$ such that for any $k ≥ 1$ ..." Oct 4 at 21:40

$$a_n=\ln n$$ 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 $$\epsilon /k$$].
• thank you! I can see how the first one implies the second now. But I don't see how $a_n = ln(n)$ satisfies the second criterion. Oct 4 at 13:07
• Use: $\ln a-\ln b=\ln (a/b)$. Also, $\ln (1+x) \le x$ . Oct 4 at 13:16