Show that if a subspace y of a metric space consists of finitely many points, then y is complete.

In order to prove that a space is complete, you have to show that every Cauchy sequence is convergent.

Now, we know that Y has finitely many points, say

And Y is a subspace of a metric space, so it inherits that metric. So, we can compute the distances among all the elements in Y: we can define the numbers

These are finitely many distances, so we can call the minimum of all these distances.

Now, a Cauchy sequence in Y is a sequence of elements in Y such that

In other words for every threshold , there exists an index such that, for all indices , the elements and are less than away.

But we know that all elements in Y can't be more than away, so if you choose , two elements are such that

So, every Cauchy sequence is eventually constant, and as such, it converges to a certain element, and thus Y is complete.

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