N neset44 Apr 2010 22 0 May 8, 2010 #1 Prove that Xn converges to X then normed Xn converges to normed X Anyone can help?

Drexel28 MHF Hall of Honor Nov 2009 4,563 1,566 Berkeley, California May 8, 2010 #2 neset44 said: Prove that Xn converges to X then normed Xn converges to normed X Anyone can help? Click to expand... 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: Spoiler \(\displaystyle |\|x_n\|-\|x\||\leqslant\|x_n-x\|\) Reactions: HallsofIvy

neset44 said: Prove that Xn converges to X then normed Xn converges to normed X Anyone can help? Click to expand... 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: Spoiler \(\displaystyle |\|x_n\|-\|x\||\leqslant\|x_n-x\|\)