Why Is This Set Closed in l2 Yet Lacks a Minimum Norm?

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SUMMARY

The discussion centers on the properties of closed sets within the context of square summable sequences in the l2 space. An example provided involves sequences like (1+1/1, 0, 0, ...) and (0, 1+1/2, 0, ...), which are isolated points. Despite being a closed set, this collection lacks an element with a minimum norm, raising questions about the nature of closed sets in functional analysis.

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Hellou!

I have a question regarding the square summable sequences:
I should find an example of a closed set from the square summable sequences and show that the closed set does not have an element with a min norm!
The professor mentioned an example:
(1+1/1 0 0 0...)
(0 1+1/2 0 0...)
...
(0...0 1+1/n 0...)

this set does not have an element with min norm, but why is it a closed set in l2 ?!
 
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they are all isolated points.
 

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