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I find it a bit strange that if $x_n \to x$ in Banach space $X$, then $|x_n|_X \to |x|_X$ by inverse triangle inequality... Surely that can't be right.

Am I correct in this? Does this have a name? Thanks

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up vote 3 down vote accepted

This is just the fact that the norm is a continuous function on a normed linear space.

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Some people think it's necessary to prove that it's continuous under the typical metric space definitions. In this case, you can show that: Assume $||x_n - a|| \rightarrow 0$ Then $||x_n|| - ||a|| \leq ||x_n - a||$ and also $||a|| - ||x_n|| \leq ||x_n - a||$ and so $| ||x_n|| - ||a|| | \rightarrow 0$ – muzzlator Feb 4 '13 at 15:06
And that is the correct proof. – ncmathsadist Feb 4 '13 at 21:20

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