How can one know every Cauchy sequence in a complete metric space converges? I am new to Cauchy sequences. I stumbled onto them in the process of learning what a Hilbert space is.  As I understand it, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. 
But consider the following statement: 
A metric space $X$ in which every Cauchy sequence converges to an element of $X$ is called complete. 
How can we know every Cauchy  sequence converges?

As I see it, the question is, "How can the criterion in the definition of completeness ever be verified (in principle, to say nothing of in practice), since the definition makes an assertion about arbitrary Cauchy sequences, and most spaces admit A Lot of Cauchy sequences?"
If one examines the proofs for, say, compact metric spaces or Euclidean spaces, there's a common theme: Show that every Cauchy sequence in $(X, d)$ has a convergent subsequence. It's a general fact in a metric space $(X, d)$ that a Cauchy sequence with a convergent subsequence is itself convergent in $(X, d)$.
Since a compact metric space is sequentially compact (every sequence has a convergent subsequence), a compact metric space is complete.
Since every real sequence has a monotone subsequence,[*] and every Cauchy sequence of reals is bounded (in the usual metric), and every bounded, monotone sequence of reals converges to a real limit, we conclude that every Cauchy sequence of reals has a convergent subsequence, and therefore converges.
And so forth. The point is, one doesn't explicitly inspect every Cauchy sequence; one proves that an arbitrary Cauchy sequence converges (e.g., by constructing a convergent subsequence), using nothing but the Cauchy property and properties of the "ambient" space $(X, d)$.
[*] Spivak's Calculus has a beautiful, elementary proof based on the concept of a peak point of a sequence $(x_{k})$, an index $n$ such that $x_{n+k} < x_{n}$ for all $k > 0$. If a sequence has finitely many peak points, one inductively constructs a non-decreasing subsequence; if instead there are infinitely many peak points, one constructs a strictly decreasing subsequence.
