# strange convergence in norm

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

-

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