Theorem 17.16.766E38862017F

Proof. Consider the barrier

It is a subgeodesic by proposition 17.16.2, and the bounded difference gives $w_0=u_0$ and $w_1=u_1$. Upper semicontinuity gives the joint upper limits at the endpoints. Off the common poles, $|w_ t-u_0|\leq Ct$ and $|w_ t-u_1|\leq C(1-t)$, so $w_ t$ runs from $u_0$ to $u_1$, and proposition 17.16.6 applies.

The lower bounds follow from $w_ t\leq u_ t^{\mathrm P}$, and the upper bounds from the chord estimate; together they give the asserted singularity type. Off the common poles, put $f(t)=u_ t^{\mathrm P}(x)$. For $0<s<t<1$, monotonicity of the slopes of secants of the convex function $f$ gives

The two-sided estimates at the endpoints cover the cases $s=0$ and $t=1$, which proves the Lipschitz bound on $[0,1]$. $\square$