Is Limit and Norm interchangable in Banach Spaces

Question

Suppose $X$ is a Banach space, and $\{x_n \}\subset X$. Does it then hold that $\lim \|y-x_n\|=\|y-\lim x_n \| $ ?

Answer 1

First let's recall what it means to say $\displaystyle\lim_{n\to\infty} x_n = x$ in a Banach space. It means $\displaystyle\lim_{n\to\infty}\|x_n-x\|=0$. The first limit is a limit of a sequence of members of the Banach space; the second is a limit of a sequence of non-negative numbers.

Note that $\|A\|=\|(A-B)+B\|\le \|A-B\|+\|B\|$, so that $$\|A\|-\|B\|\le\|A-B\|,$$ and in the same way show that $$\|B\|-\|A\|\le\|A-B\|,$$ so we have $$ |\, \|A\|-\|B\| \,|\le \|A-B\|. $$ Hence $$ |\,\|y-x_n\|-\|y-x\|\,| \le \|(y-x_n)-(y-x)\| = \|x_n-x\| \tag 1 $$ Consequently, if the rightmost term in $(1)$ goes to $0$, then so does the leftmost term.