Theorem: Let (X, norm) be a m-dimensional normed linear space. Let {xn} and x be expressed by Xn=λn1*e1+...+λnm*em and X=λ1*e1+...+λm*em. Then xn→x IFF λnk→λk for k=1,...,m.(adsbygoogle = window.adsbygoogle || []).push({});

The proof for λnk→λk for k=1,...,m implies xn→x is rather straight forward. I am having trouble with the proof for λnk→λk implies xn→x.

Here is the proof from class lectures.

View attachment 43442

This is a proof by contradiction.

First, since the coordinates do not converge to 0, then at least one set of λnj's do not converge to 0.

Then a subsequence is built with each element greater than some positive number, r, which must exist since at least one of the λnj's do not converge to 0.

Observe that each element of this subsequence is bounded below by r and above by the greatest λnk, called Mi.

Construct yi as xni/Mi. Then yi→0 because xni→0.

Then I get lost.

1. I am not sure why |μik|<=1.

2. I am not sure why sum(|μik|)>=1.

Can anyone help me fill in these missing parts? I've worked through this proof for over an hour and haven't gotten any further.

Thanks in advance,

Scott

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# Proof: Norms converge imply Coordinates converge

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