Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$.
Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to \mathbb{R}$, denoted by $$f_n(x) = \begin{cases} 0 & x \in [-1, 0],\\ nx & x \in [0, \frac{1}{n}],\\ 1 & x \in [\frac{1}{n}, 1] \end{cases}.$$
Show that the sequence $(f_n)$ does not converge in $C[-1, 1]$.
For this problem, I have shown that if $f_n$ converges to $f$, then $f$ has to be $0$ for $x \in [-1, 0]$ and $f$ has to be $1$ for large $x$. Then what should I do?