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rfvelasco
12.07.2019 •
Mathematics
Show that if a subspace y of a metric space consists of finitely many points, then y is complete.
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Ответ:
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.
Ответ:
Step-by-step explanation:
there is no equation