I was reading about Cantor function but I don't really understand the reasonning from wikipedia.
If we have
$\max_\limits{x \in [0, 1]} |f_{n+1}(x) - f_n(x)| \le \frac 1 2 \, \max\limits_{x \in [0, 1]} |f_{n}(x) - f_{n-1}(x)|, \quad n \ge 1$
Then we have
$\max\limits_{x \in [0, 1]} |f_{n+1}(x) - f_n(x)| \le \frac {1} {2^{n+1}} \max\limits_{x \in [0, 1]} |f_{1}(x) - f_0(x)|$
So $\forall x, f_n(x)$ is a cauchy sequence and since $\mathbb{R}$ is complete, we have that $f_n(x)->f(x)$ pointwise. But am missing the argument that allows us to say that
$\max\limits_{x\in[0,1]}|f_n(x)-f(x)|\le\frac{1}{2^{n+1}}\max\limits_{x\in[0,1]}|f_1(x)-f_0(x)|$