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.

damm bro the struggle

step-by-step explanation:

fist on the left goes with the third on the right. second on the left goes with the fourth on the right. third on the left goes with the second on the right. the fourth on the left goes with first on the right