Why is the Cauchy convergence theorem important? I mean, what other concepts are related to it? Are there practical uses for the Cauchy convergence theorem?
The "Cauchy Convergence Theorem", also called the "Cauchy Criterion", says that if a sequence of real numbers is a "Cauchy sequence" ( , with m and n going to infinity independently) then it converges.
One nice thing about this property is that it does not require that you know the limit in order to prove that a sequence converges, as you would if you used the definition of "limit of a sequence".
But the real reason it is important is that it is a "defining" property of the real number system. That is, it is NOT true of the rational numbers- the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, where each number is one more digit in the decimal expansion of is a Cauchy sequece of rational numbers but does not converge to a rational number.
You could take all of the axioms defining the rational numbers and add the "Cauchy Criterion" as an axiom and have the real numbers.
In fact, one way to define the real number system is this: Define two Cauchy sequences of rational numbers, and to be "equivalent" if and only if and define the real numbers to be the equivalence classes of such sequences under that equivalence relation. In that sense, would be defined as the equivalence class that contains, among other sequences, the sequence I gave above.
There are a number of "defining" properties of the real numbers-
The least upper bound property- if a set of real numbers has an upper bound then it has a least upper bound (and, of course, if a set of real numbers has a lower bound then it has a greatest lower bound).
Monotone convergence- if an increasing sequence has an upper bound then it converges (and, of course, if a decreasing sequence has a lower bound then it converges).
Every bounded sequence has a convergent subsequence.
Every closed and bounded set of real numbers, with the usual topology, is compact.
The set of real numbers, with the usual topology, is connected.
Those are all equivalent to one another and the Cauchy Criterion- given any one of them, you can prove the others.