Proof for banach Space 1

Apr 2010
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Prove that Xn converges to X then normed Xn converges to normed X

Anyone can help?
 

Drexel28

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Prove that Xn converges to X then normed Xn converges to normed X

Anyone can help?
This is horrible notation. I assume your saying that \(\displaystyle \left\{x_n\right\}\) is a sequence in a Banach space \(\displaystyle B\) and \(\displaystyle x_n\to x\) and to prove that it follows that \(\displaystyle \|x_n\|\to\|x\|\), right?

Hint:

\(\displaystyle |\|x_n\|-\|x\||\leqslant\|x_n-x\|\)
 
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