Using the sequence definitions, if $X$ is compact, prove that $X$ is complete.
The definition of "compact" to be used is:
$(X,\ d)$ is compact iff every sequence in $X$ has a subsequence that converges to a point in $X$.
Also,
$(X,\ d)$ is complete iff every Cauchy sequence in $X$ converges to a point in $X$.
My attempt:
Suppose $\{x_n\}$ is a Cauchy sequence in $X$. We need to prove that $\{x_n\}$ converges to a point in $X$.
Since $X$ is compact, $\{x_n\}$ has a subsequence $\{x_{n_i}\}$ that converges to a point $x\in X$. We need to prove that $x_n\to x$ but I don't know how to proceed.
PS: Please re-open this question. I'm trying to prove this using the sequence definitions only, not using the Heine-Borel theorem.