If Q-Cauchy sequences must limit to a rational, how can they construct the reals? I'm currently in a graduate Math for Economists course, and we spent last week learning how to construct the reals from the rationals using Cauchy sequences and their equivalence classes. I know that I'm missing something, because from the definitions presented to me, I fail to see how the Q-Cauchy sequences can represent the irrationals.
We define each real number,r, as the equivalence class of Cauchy sequences with r as their limit. Yet, without the reals, we only have Q convergence. So a sequence Q-Converges if for all $n>N, |a_n-L|< E,$ where $L,E$ are rational.
So, how can we define $\pi$, or $\sqrt{2}$? There is no Q-Cauchy sequence which Q-converges to anything other than a rational.

I struggled with this exact idea when I first learned the reals-from-rationals construction.  Here's the example that made me understand.
Every irrational number has an (infinite) decimal representation.  Construct that particular irrational by enumerating the decimal representation, one digit at a time.  For example, the sequence that converges to $\pi$ is:
$3$
$3.1$
$3.14$
$3.141$
$3.1415$
$3.14159$
and so on.  Each of these terms is clearly rational ($3.14159 = \frac{314159}{100000}$), but the limit is clearly $\pi$.
