Convergence in $C([0, 1])$

Question

Let's consider the space of continuous functions on the interval $[0, 1]$. We'll denote it by $C([0, 1])$.
Now we can define a sequence: $$f_n(x) = x(1-x^n).$$ It's easy to find it's pointwise limit: $$\lim_{n \to \infty} f_n(x) = g(x) = \begin{cases} x,\text{if } x\in[0, 1)\\0, \text{if } x = 1\end{cases}$$ I would like to show that the sequence does not converge in norm. $$\left\lVert f_n- g \right\rVert = \max_{x \in [0,1]} |f_n(x)-g(x)|$$ I don't know where to go from here. I would appreciate any tips or hints.

Answer 1

Notice that for $x=1,$ $|f_n(x) - g(x)| = 0$. Hence $$ \sup_{x\in[0,1]}|f_n(x)-g(x) | = \sup_{x\in[0,1)}|f_n(x)-g(x)| = \sup_{x\in[0,1)} |x(1-x^n)-x| = \sup_{x\in[0,1)} |x^n| = 1, $$ for all $n$, so $f_n \not\to g$ uniformly.