Pointwise closure of the bounded Lipschitz unit ball ($f_i\to f\text{ pointwise},\Vert f_i{\Vert }_{\mathrm{BL}}\le 1⟹\Vert f{\Vert }_{\mathrm{BL}}\le 1$)

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.