By definition of Cauchy sequence, if $(a_n)$ is a Cauchy sequence, then we have $|a_n-a_m| \rightarrow 0$, as $n,m \rightarrow \infty$.
Suppose $(b_n)$ is a sequence. For any $m\in \mathbb{N}$, we have $|b_n-b_m|\rightarrow 0$ as $n\rightarrow \infty$. Does this imply that $(b_n)$ is a Cauchy sequence?
My first thought was no, because we want $n,m$ to grow at the same time. But now I'm confused on the "for any $m$" part.
Thanks!
Since these are definitions you decide whether $m$ and $n$ grow at the same time or not. The definition of Cauchy sequence requires so. Your definition does not. (Keep in mind that Cauchy's definition is not inherently better or valuable than the one you gave, simply Cauchy's "happens" to have many interesting properties) What might have confused you is the $n,m \rightarrow \infty$ part, since it does not specify the fact that n,m go to infinity together. A clearer way of writing it would be: $n,m > k$ with $k \rightarrow +\infty$. The interpretation of your definition would instead be: "Let's fix $b_m$ for a specific value $m$, then send $n$ to infinity. Then $|b_n-b_m| \rightarrow \infty$. This applies to every possible choice of $m$". As mentioned by infinitylord, the only sequence that satisfies the property is the constant one (which indeed is Cauchy).