description/proof of that for metric space induced by normed vectors space, if for each sequence s.t. series of norms converges, series converges, space is complete
Topics
About: vectors space
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of normed vectors space.
- The reader knows a definition of metric induced by norm on real or complex vectors space.
- The reader knows a definition of complete metric space.
- The reader admits the proposition that for any Cauchy sequence on any metric space, if a subsequence of it converges to a point, the sequence converges to the same point.
Target Context
- The reader will have a description and a proof of the proposition that for the metric space induced by any normed vectors space, if for each sequence such that the series of the norms of the terms of the sequence converges, the series of the terms of the sequence converges, the space is complete.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
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Statements:
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2: Proof
Whole Strategy: Step 1: take any Cauchy sequence,
Step 1:
Let
Step 2:
For each
Let us take
Step 3:
Let us see that
Step 4:
By the supposition of this proposition,
But
Step 5:
As
So, as each Cauchy sequence converges,