Completenees of $\mathbb{R}$ via Bolzano-Weierstrass theorem- Let me understand if i understood correctly

Question

I know that a metric space is complete if every Cauchy sequence converges to an element of the space (we can call it $X$).
To prove that the metric space is complete, let $\left(x_{n}\right)$ be any Cauchy sequence.
I know thanks to the Th. of Bolzano-Weierstrass that every bounded sequence of real numbers has a convergent subsequence, so $\left(x_{n}\right)$ contains a subsequence converging to some point $a \in X$.
But then what proposition make the whole sequence $\left(x_{n}\right)$ converges to $a$ to show that $X$ is complete?