I have a basic problem understanding the convergence of the "tent" function
\begin{equation} x_n(r) = \begin{cases} nr,\; r \in \left[0, \frac{1}{n}\right]\\ 2 - nr, \; r \in \left[\frac{1}{n}, \frac{2}{n}\right]\\ 0,\; r\in \left[\frac{2}{n}, 1\right] \end{cases} \end{equation}
to the zero function, $x_0(r)=0, \; \forall r\in [0, 1]$ pointwise in $r$.
I understand that the function concentrates around zero in the ordinates that depend on $n$, but doesn't it still place a lot of mass at that point?
Also, why is the convergence not uniform using the $\sup$-metric?