Norm equivalence and the sequence limit of normalized vectors

Supposedly $v\ne0$. Let $u_k=v_k/\|v_k\|_1$ and $u=v/\|v\|_1\ne0$. By assumption, $u_k\to u$ (with respect to both norms, because all norms are equivalent) and hence $\|u_k\|_2\to \|u\|_2$ (because the norm function is continuous). Therefore $$ \left\|\frac{u_k}{\|u_k\|_2}-\frac{u}{\|u\|_2}\right\|_2 \le \frac{1}{\|u_k\|_2}\|u_k-u\|_2 +\left|\frac{1}{\|u_k\|_2}-\frac{1}{\|u\|_2}\right|\|u\|_2 \to0. $$ Consequently, $v_k/\|v_k\|_2=u_k/\|u_k\|_2\to u/\|u\|_2=v/\|v\|_2$.