I don't understand how this piecewise converges to $0$
Determine the point wise limit of $(f_n)$ on the indicated interval, and decide whether $(f_n)$ converges uniformly to this function
$f_n = \left\{\begin{matrix} 0 & x \leq n \\ x-n & x \geq n \end{matrix}\right.$ on $[a,b]$, and on $\mathbb{R}$
So since $n \geq 0$, I only need to look at $x \geq 0$. For $[a,b]$, as $n\to\infty$, $f_n$ doesn't seem to have a pointwise limit and on $\mathbb{R}$, it also doesn't seem to have a limit as it becomes "periodic"
The answer book says it converges to $0$ and it says it doesn't converge uniformly on $\mathbb{R}$, but it does converge uniformly on $[a,b]$. Why?