Recently I found a supposed alternate definition for a Cauchy sequence in $\mathbb{R}^n$ given by:
A sequence is a Cauchy sequence iff given a radius $r$ and an index $N$, the whole sequence starting from $x_N$ is in a ball with such radius, that is, $\{x_i\}_{i = N}^\infty \in B(c, r)$, where $c$ is the ball's center.
I'm having a lot of trouble proving that. Do I need to say that $c$ is the limit of the sequence? Doesn't this already implies that the sequence is Cauchy, as every convergent sequence is a Cauchy sequence in $\mathbb{R}^n$?
You don't even need to use the fact that there is convergence happening, however your statement should be:
Note that the order of quantifiers is slightly different than what you wrote.
Here's how you can show equivalence:
If $\{x_k\}$ is Cauchy, then for each $r > 0$, there is $N$ large such that: $$\sup_{n, m \geq N} |x_n - x_{m}| \leq r. $$ In particular, if $m \geq N$, then $|x_n - x_N| \leq r$, and so $\{x_n\}_{n \geq N} \subset B(x_N, r)$.
On the other hand, if the conclusion holds then let $\epsilon > 0$. We have that $\{x_n\}_{n \geq N} \subset B(c, \tfrac{\epsilon}{2})$, for some $c$. But now note that this implies: for each $n, m \geq N$: $$ |x_n - x_m| \leq |x_n - c| + |x_m - c| \leq \epsilon. $$ And so $\{x_k\}$ is Cauchy.