Given $f:[0,1]\rightarrow\mathbb{R}$ given by:
$$f(x)= \begin{cases} 2x & x\in[0,\frac{1}{2}]\\ -2x+2 & x\in[\frac{1}{2},1] \end{cases} $$
How can I construct a sequence of $C^\infty$ functions that are always smaller but converge to $f$? Is there a systematic way to do it?
I'm trying to have all the details for the proof that the space of $C^1$ functions is not a Banach space in the uniform convergence norm.
Consider the family of hyperbolas
$$(y-1)^2 - (2x-1)^2 = r^2$$
In the limit as $r\to 0$, this approaches the mutual asymptotes of all of the hyperbolas
$$|y-1| = |2x-1|$$
So consider the sequence of functions
$$f_n(x) = 1 - \sqrt{\frac{1}{n}+(2x-1)^2}$$
I will leave the proof of smoothness on the interval $[0,1]$ to you.