the definition of irrational numbers and the proof of Cauchy's theorem At page 281 of The History of the Calculus and its Conceptual Development

Why the necessity of the condition here doesn't require a previous definition of the system
of real numbers, especially the definition of irrational numbers, but the proof of the
sufficiency of the condition does ?
 A: Consider any metric space $(X,d)$. Let $\epsilon >0. $If $(x_n)\subset X$ converges to $x\in X$ then $\exists N\in\mathbb{N}$ such that $\forall n>N, d(x_n,x)<\frac{\epsilon}{2}.$ Now let $m,n\in\mathbb{N}$ be such that $m,n>N$. Invoking triangle inequality you prove that $d(x_n,x_m)<\epsilon$, whence $(x_n)$ is Cauchy. 
This result holds true for any metric space. However the convergence of a Cauchy sequence to a point in the given metric space depends on the space. In fact metric spaces in which all Cauchy sequences are convergent are called complete metric spaces. Now the space $\mathbb{Q}$ is not complete. But $\mathbb{R}$ is complete. Note that $\mathbb{R}=\mathbb{Q}\bigcup(\mathbb{R-Q})$ where $\mathbb{R-Q}$ is the set of irrationals defined accordingly. I think this is what the author meant.
A: Note that the concepts of limit for functions and sequences can be defined over the system of rational numbers also. If you see the definitions of these concepts they are only based on density property of the numbers in question. Both systems of rationals and reals have density property, both have order relations and both are fields. There is no difference between them apart from the completeness property.
It is completeness of real numbers which ensures that many interesting limits exist. Thus based on definitions we can prove that every sequence (in any system either rational or real) which has limit (in the same system) also satisfies the Cauchy's criterion. But to prove the converse (which is related to existence of limits) we must use completeness of real numbers. In the system of rational numbers we do have Cauchy sequences which have no limit.
To convince yourself, you should prove that the sequence $\{x_{n}\}$ defined by $$x_{1}=1,x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{2}{x_{n}}\right)$$ is a sequence of rational numbers which satisfies Cauchy's criterion, but there is no rational number to which this sequence converges. It is this inadequacy of rationals (which BTW is very technical in nature compared to the other inadequacy which Dedekind observed) which led Cantor to develop the theory of real numbers as Cauchy sequences of rationals.

Now that we have sound theories of real numbers at our disposal no one really bothers to ponder over the need for real/irrational numbers in mathematics. But throughout history mathematicians felt that there was something deeply lacking in the rational numbers which could not be supplied by algebra (that is by adding algebraic numbers to the field of rationals). Most mathematicians took the shortcut and assumed the existence of real numbers with the desired property to fix what was lacking in the rationals. All this was based on intuition and only in nineteenth century came rigorous theories of real numbers which put them on a sound foundation.
And it seems that history repeats itself. Modern analysis textbooks and curriculum again takes real numbers for granted (the axiomatic approach to real numbers) and the students are almost never aware of the need / importance of real numbers. And those theories of real numbers are thought of as just some set theoretic stuff on par with Peano's axioms.
A: It is very easy to show that if $x_n \to x$ then $(x_n)_n$ is a Cauchy sequence directly. 
To show that if $(x_n)_n$ is Cauchy, then $x_n \to x$ for some $x$ is a bit harder. If the $x_n$'s were rational and we were seeking a rational $x$, then this would be impossible. It follows that we need to work with irrational numbers as well. 
