If I want to show that something is Cauchy, should I show it converges and then show it is Cauchy or should I go at it straight from the definition. I am just trying to figure out generally what to do for a sequence.
I don't have a concrete example. I am sorry, but like if the question was Prove that this series is Cauchy... How would I go about doing it?
On
If you have a candidate for what the sequence converges to, then it can be easier to show that the sequence converges. If you don't, though, it can be hard to show that a sequence converges just from the definition. In that case it can be easier to use the definition of Cauchy
On
If a sequence does converge, then it is a Cauchy sequence. Be careful, though, that not always does a Cauchy sequence converges. For exemple, the sequence $x_n = \frac{1}{n}$ is a Cauchy sequence, but does not converge in $]0,1]$. It does converge in $[0,1]$. Search about completeness of a topological space (or a metric space, for what it's worth).
As a general rule:
If you know what it converges to, i.e. if what it converges to is obvious, it is often easier to check convergence directly. For instance, the sequence $1 - \frac{1}{n}$ obviously converges to $1$, and it's easier to check this than it is to check the Cauchy criterion.
If you have no idea what it converges to, the Cauchy criterion is a valuable tool. This is for example the most elementary way to show the partial sums of a series converge.