I'm having trouble constructing the Hausdorff completion of a uniform space $(X,U)$ using pseudometrics. I know that every uniformity on a space $X$ is made by pseudometrics. Here is my idea:
Let $(X,U)$ be a uniformity generated by pseudometrics $(d_i: X \rightarrow\mathbb{R}_{\geq 0})_{i \in I}$. Define 'Generalized' Cauchy Sequences (GCS) as followed: $(x_n)_{n \in \mathbb{N}}$ is a GCS iff $$\forall \varepsilon > 0 \forall K \subset I \text{ finite }\forall i \in K \exists n_0 \forall p,q \geq n_0 : d_i(x_p,x_q)<\varepsilon$$ Let $\mathcal{C}$ be the set of all GCS and define a equivalence relation as followed: $$ (x_n)_{n \in \mathbb{N}}R(y_n)_{n \in \mathbb{N}} \Leftrightarrow \forall \varepsilon > 0 \forall K \subset I \text{ finite }\forall i \in K \exists n_0 \forall n \geq n_0 : d_i(x_n,y_q)<\varepsilon$$
Let $X'$ be the set $\mathcal{C}/R$. Define pseudometrics $d'_i((x_n)_{n \in \mathbb{N}},(y_n)_{n \in \mathbb{N}})$ as $\lim_{n \rightarrow \infty} d_i(x_n,y_n)$. Let $k: X \rightarrow X'$ be the canonical function.
Now, I can prove that $X'$ is Hausdorff (thanks to the equivalence relation), $k(X)$ is dense is $X'$ (every point in $X'$ is the limit of images of points in $X$), but I'm having trouble proving that $X'$ is complete.
Basically, I have 2 questions: is what I'm doing correct (and if so, how do I proceed) or else, what am I doing wrong here (and how to correct it).
As always, any help would be appreciated.