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Vollständigkeit in endlich dimensionalen Vektorräumen

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Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise

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Level 5 -
Level 6 -

Exercise

https://texercises.com/exercise/vollstandigkeit-in-endlich-dimensionalen-vektorraumen/

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Exercise:
Sei dgeq . Jede abgeschlossene Teilmenge von mathbbR^d ist vollständig. Insbesondere ist mathbbR^d vollständig.

Solution:
Beweis. Man zeigt zuerst dass mathbbR^d vollständig ist. Sei a_n_n eine Cauchy-Folge im mathbbR^d und sei pi_ja_n für nin mathbbN und jin ...d die j-te Komponente von a_n. Für epsilon fixiert sei Nin mathbbN s.d. ||a_m-a_n|| epsilon für alle nm geq N. Insbesondere gilt |pi_ja_m-pi_ja_n| epsilon für alle jin ...d und nmgeq N womit die Folge der Komponenten pi_ja_n_n eine Cauchy-Folge in mathbbR ist. Nach Satz . ist mathbbR vollständig womit die Cauchy-Folge pi_ja_n_n für jedes jin ...d konvergent sein muss. Nach Proposition . ist also auch a_n_n konvergent was impliziert dass mathbbR^d vollständig ist. Sei nun Asubseteq mathbbR^d eine abgeschlossene Teilmenge und sei a_n_n eine Cauchy-Folge in A. Dann besitzt die Folge a_n_n einen Grenzwert bin mathbbR^d. Nach Lemma . muss aber b in A liegen womit die Cauchy-Folge a_n_n also in A konvergiert. Somit ist A vollständig.

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Tags	analysis, beweis, eth, fs23, proof, topologie, vollständigkeit
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Difficulty	(3, default)
Points	0 (default)
Language	(Deutsch)
Type	Proof
Creator	rk
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