Can anyone explain the reasoning behind this particular line ? $$ \mathop {\sup \,|f_n }\limits_{x \in [0,1]} (x) - f(x)| = \mathop {\sup \,| - \frac{{2x}}{n}}\limits_{x \in [0,1]} + \frac{1}{{n^2 }}| $$
And where do the values ${ - \frac{{2x}}{n}}, \frac{1}{{n^2 }}$ come from ? I tried reading about uniform convergence from various materials but I really don't get the reasoning behind the proof. If anyone could shed some insight I would be most grateful.
$$\left|f_n(x)-f(x)\right|=\left|\left(x-\frac1n\right)^2-x^2\right|=\left|x^2-\frac {2x}{n}+\frac{1}{n^2}-x^2\right| \\ =\left|-\frac {2x}{n}+\frac{1}{n^2}\right|\le \left|-\frac {2x}{n}\right|+\left|\frac{1}{n^2}\right|\le \frac {2}{n}+\frac{1}{n^2}, \forall x\in [0,1],\forall n\in \mathbb{N}. $$