Let (xm) be a Cauchy sequence in a normed linear space x of dimension n. Let {x1,x2,...,xn} be a basis for X. Each term x''' has a unique representation

1. Using lemma 1.1, show that each sequence of scalars xmi is a Cauchy sequence in „œ and hence converges to some αi ˆˆ „œ.
2. Define

Show that x ˆˆ X and that xm †’ x.
3. Conclude that every finite-dimensional normed linear space is complete.

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