Let $(V,\Vert\cdot\Vert)$ be a normed vector space and suppose $\{u_n\}_{n=1}^\infty$ is a sequence in $V$ that converges to $u$. If each term in $\{u_n\}_{n=1}^\infty$ has unit length, can we conclude that $u$is of unit length?
I would say it is true, because for all $u,w$ in $V$, we have$$\lvert\Vert u\Vert-\Vert w\Vert\rvert\leq\Vert u+w\Vert\leq\Vert u\Vert+\Vert w\Vert,$$and for each $n$, we have$$\lvert\Vert u-u_n\Vert-1\rvert=\lvert\Vert u-u_n\Vert-\Vert u_n\Vert\rvert\leq\Vert u\Vert\leq\Vert u-u_n\Vert+\Vert u_n\Vert=\Vert u-u_n\Vert+1.$$Applying the sandwich theorem leads to $\Vert u\Vert=1$. Am I right? Thank you.