Originally Posted by

**Ares_D1** I know what you mean. I *think* what they're trying to emphasize is that you can't just prove convergence and then simply state "Since the sequence is convergent, it necessarily follows (from Theorem 31) that the sequence is Cauchy." This is a perfectly valid statement, of course, but obviously they want us to take it a step further. So, given that we know the sequence is convergent, we can use this to show that for any $\displaystyle \epsilon > 0$, there exists some $\displaystyle N \in \mathbb{N}$ such that for all $\displaystyle m, n \geq N$, we have

$\displaystyle |a_m - a_n| < \epsilon$

Which *is* the definition of a Cauchy sequence. Again, this is essentially identical to the method used in Theorem 31, adapted to this particular sequence.