# Defn of Completeness and Convergence in Norm

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...

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If you know $X$ is complete that surely $x$ must belong to $X$, otherwise it is easy to construct a counterexample e.g. by looking at rational numbers converging to some none rational number. – AD. Mar 8 '12 at 7:37
I reference to the proof you mention might help. – AD. Mar 8 '12 at 7:38