Suppose $x_n$ is Cauchy, then $\lim\sup x_n = \lim\inf x_n$.
Can't use the fact that all Cauchy sequences converge, as we haven't shown this yet. Here is my attempt.
Since $x_n$ is Cauchy, then $x_n$ is bounded.
First define $b_n = \sup\{x_n, x_{n+1}, ...\}$. Note that this sequence is decreasing and bounded below. Thus $\lim b_n = \lim\sup x_n \leq \inf\{x_n, x_{n+1}, ...\}$.
Take $a_n = \inf\{x_n, x_{n+1}, ...\}$. Note that this sequence is increasing and bounded above. Thus $\lim a_n = \lim\inf x_n \geq \sup\{x_n, x_{n+1}, ...\}$.
Since $\lim\sup x_n \leq \inf\{x_n, x_{n+1}, ...\} \leq \sup\{x_n, x_{n+1}, ...\} \leq \lim\inf x_n$. But $\lim\sup x_n \geq \lim\inf x_n.$
Thus $\lim\sup x_n = \lim\inf x_n$
I don't think this is the right approach, because I didn't even use Cauchy property.