- #1
geoffrey159
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Homework Statement
Show that finite dimensional normed vector spaces are complete.
Homework Equations
##E## is a finite dimensional vector space and ##N## a norm on ##E##
The Attempt at a Solution
If ##\{x_n\}_n## is a Cauchy sequence in ##(E,N)##, then it is bounded and each term of the Cauchy sequence belongs to a closed ball of ##E## centered in 0 with radius ##r = \text{sup}_n\{N(x_n)\}##. This ball is compact in finite dimension, so ##\{x_n\}_n## has at least an adherence value in this ball, and a Cauchy sequence that has an adherence value converges to this value. Which proves ##E## is complete. Is it correct ?