If we allow the metric to be $d(x,y)=|x-y|$, we must prove that this is complete. Now, I have proven all properties of a metric space. However, I don't particularly now where to begin to prove that this is complete. 

I understand that we need every Cauchy sequence in this space to converge inside the space itself. Now, I know the definition of Cauchy convergence, but I don't particularly know how to even begin by showing that this is complete. I have much harder metrics to prove but I would like to try to understand this simpler space before I move onto more difficult ones.

Since we know that the output of this distance will always be some constant in $\mathbb{R}$, and since we are working with $\mathbb{R}$, then it must always converge to some real number, and thus its complete?