Question 7.E.3.15: Explain why the function m(v) used in the development of the......

Consider

x_n = \begin{pmatrix}1  −  1/n\\ -1\end{pmatrix} → x = \begin{pmatrix}1\\-1\end{pmatrix},

but m(x_n) = −1 for all n = 1, 2, . . . , and m(x) = 1, so m(x_n)  \cancel{→}  m(x). Nevertheless, if lim_{n→∞}  x_n ≠ 0, then lim_{n→∞}  m (x_n) ≠ 0 because the function \tilde{m}(v) = |m(v)| = ||v||_∞ is continuous.