Defn of Completeness and Convergence in Norm

Question

Let $X$ be a normed space. We say that $X$ is complete if every cauchy seq in $X$ converges to an element of $X$ in norm. Now, in proofs of completeness, we start with $\{x_n \}$ Cauchy, and it seems we should first exhibit an element $x$ of $X$ and show that $\{x_n \}$ converges to $x$ in norm. But, I read a proof where first, they constructed the limiting element $x$, and then showed that $\{ x_n\}$ tend to $x$ under the norm, and used that convergence result to argue that $x$ was actually a member of the space $X$. I'm not sure why I find this argument fishy. I feel you can't say something converges in norm if the limiting element doesn't even exist in your space until after it converges...