I know that every completion is a closure of a metric space, since every convergent sequence is cauchy and and the limit of that sequence will exist within the completion.
At the same time, from my understanding, every cauchy sequence will bunch closer together and get arbitrarily close to something, but it is just a question as to whether or not that element it gets closer to actually exists in the space.
This leads me to the question as to whether every closure of a metric space is a completion, because we would just be adding the limits to sequences which exist outside of the original space, including the limits of nonconvergent cauchy sequences.
So is there an example of a closure which is not a completion? Or are these notions equivalent?