Pointwise closure of the bounded Lipschitz unit ball
= Pointwise closure of the bounded Lipschitz unit ball
{title2=$f_i\to f\text{ pointwise},\quad\|f_i\|_{\rm BL}\leq1\ \Longrightarrow\ \|f\|_{\rm BL}\leq1$}
For every fixed $x\ne y$ and $z$, the sum $|f_i(x)-f_i(y)|/d(x,y)+|f_i(z)|$ is at most one. <Pointwise convergence> preserves this inequality. Taking the independent suprema over the pair $(x,y)$ and the point $z$ proves the assertion. Taking each bound separately would only give the weaker estimate two.