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?

You have your statement wrong: 
A metric space $X$ in which every Cauchy sequence that has a limit converges to an element of $X$ is called complete.
And you sense of what "Cauchy" means is a bit vague. For example, in the space $\Bbb{R}$ (the real line), the sequence $H_n \equiv \sum_{i=1}^n \frac{1}{i}$ has arbitrarily small distances between successive members, but is not is Cauchy because there for any given $m$ there is an $H_m$ with $m>n$ that is arbitrarily far from $H_m$.  So this sequence it is not considered in deciding whether $\Bbb{R}$ is a complete space.
One way to prove that all Cauchy sequences that have a limit converge to an element of $X$ is to show that $X$ is compact, for that is a property of a compact space.  This is easy to see:  If $X$ is compact, then it contains all its limit points, and the limit of a Cauchy sequence is a limit point.  
It turns out that compactness is a bit of a stronger condition than is needed for completeness. For example, $\Bbb{R}$ is not compact. But since it does contain open sets including each of the points it contains, it is easy to see that $\Bbb{R}$ contains all its limit points, and therefore the limit of a Cauchy sequence of elements of $\Bbb{R}$ is in $\Bbb{R}$ so $\Bbb{R}$ must be complete.
I will admit that since you generally can't just test every possible Cauchy sequence, there are some spaces for which it is tough to prove whether they are complete or not. Sometimes very clever arguments are needed.
