Given that we have sequence of piece-wise functions $S_{n}$ on [0,2] given by $ \frac{1}{n}x+x^{2}$ for ,$ 0\leq x \leq 1$ and $\frac{1}{2n}$ for , $1<x \leq 2$
where sup metric distance between functions $ d_{s}(h,g) = \underset{x \in {[0,2]}}{sup} | f(x) -g(x) |$
How can we find the expression for the sup distance between any two functions for any arbitrary n under sup metric.
Hint
$$\sup_{x\in[0,2]} |f_n(x)-f_m(x)|=\max\Bigg\{\sup_{x\in[0,1]} |f_n(x)-f_m(x)|,\sup_{x\in(1,2]} |f_n(x)-f_m(x)|\Bigg\}$$where $f_n$ and $f_m$ are two arbitrary functions for $m,n\in\Bbb N$.