What is the completion of a metric space  $(\mathbb{Q}, |\ \ |)$? What is the completion of a metric space $(\mathbb{Q}, |\ \ |)$?
 A: A metric space $X$ is complete if every Cauchy sequence $a_n \in X$  converges to an element $a \in X$.
The completion of $X$ therefore is the metric space $\bar X$, that contains all elements of $X$, plus the limits of all possible cauchy sequences in $X$ equipped with the same metric as $X$. There is no straight forward way in finding the completion of a Metric space.
In your particular case it was already mentioned in the comments, that the completion of $\mathbb{Q}$ with the canonical metric is $\mathbb{R}$ with the canonical metric. 
A: If $X$ is a metric space, then the completion of $X$, denote it by $X_c$, is the smallest complete metric space containing $X$ as a subspace. That is, if $Y$ is complete and contains $X$ as a subspace, then $Y$ also contains $X_c$ as a subspace. 
If $Y$ is complete, then a subspace of $Y$ is complete if and only if it is closed in $Y$. 
Assuming that you have proved these two things, you can use them to find that the completion of $(\mathbb{Q}, |\cdot|)$ is $(\mathbb{R}, |\cdot|)$. What must a complete subspace of $\mathbb{R}$ containing $\mathbb{Q}$ be?
