How can some function $f_n(x)$ is point wise bounded but not uniformly bounded

Question

let $f_n(x)≤ M(x)$ so that $f_n(x)$ is point wise bounded. Now let $M = max (M(x))$ then $f_n(x) ≤ M$ for all x. Wouldn't this make every point wise bounded functions of the form $f_n(x)$ uniformly bounded? We can just take the maximum and bound them all together.

How come some functions are point wise bounded but not uniformly bounded then?

Answer 1

Consider $f_n(x) = x$. ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$

Answer 2

The fact is that the $\max(M(x))$ might not exists and you have to switch to a the $\sup$ which can be $+\infty$.