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Answered Same Day Apr 28, 2021

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Rajeswari answered on Apr 30 2021

132 Votes

1) To prove that every compact metric space is complete
A topological space is compact if each of its over cover has a finite subcover. In other words, X is compact if for C, all possible collections of open subsets of X, there is a finite subset F of C such that
X= U x (i.e. X is union of all subsets of x where x belongs to F.
A metric space (X,d) is complete if and only if every collection {Fn} of non-empty closed sets with
And diam Fn 0,
Since we have (X,d) a compact metric space we have
is non empty.
Since diam Fn≤diam Fm 0, when m tends to infinity.
This implies contains atmost one point.
i.e. is singleton set.
It follows that the metric space is complete.
2) Given a Cauchy sequence {xn}, select an epsilon >0 such that
Now for
Hence we have trapped all the term including and after the kth, in a ball of size epsilon around the term x_k.
For the terms before k, we have only a finite number of terms as
If possible let atleast one of them be far away from x_k with distance d>epsilon.
Let us consider then everything in the sequence must lie inside for some
i.e. every...

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