Consider the sequence of functions $f_{n}:[0,1] ightarrow \mathbb{R} $ given by $$ f_{n} (x)=\left\{\begin{array}{11} 1-n x & \text { if } 0 \leq x \leq 1 / n l 1 & \text { otherwise } \end{array} ight. $$ for $n>0$, which converges pointwise to $f(x)=1$ as $n ightarrow \infty$. Show that $\left\{f_{n} ight\}_{n=1}^{\infty}$ does not converge to $f$ in the uniform norm, but it does converge using the norm defined in Problem (1). (As a consequence, for infinite dimensional vector spaces, there are norms that are not equivalent.) CS.VS. 1441