Proof of Increment Bound for a Convex Function through an Extended Point

Since $x$ and $z$ lie in $D$ and hence in $C$, and since $0\le\lambda$ and $\lambda\le1$, Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$ applies to the representation $y=\lambda z+(1-\lambda)x$ and gives

$$u(y)\le\lambda\,u(z)+(1-\lambda)\,u(x).$$

Adding $-u(x)$ to both sides, by claim 3 of Elementary Order Arithmetic in an Ordered Field, and simplifying the right-hand side by the field axioms,

$$u(y)-u(x)\le\lambda\,u(z)+(1-\lambda)\,u(x)-u(x)=\lambda\bigl(u(z)-u(x)\bigr).$$

Since $z\in D$ we have $u(z)\le M$, and since $x\in D$ we have $m\le u(x)$, so adding $-u(x)$ to the first inequality and then $M$ to the second, again by claim 3 of Elementary Order Arithmetic in an Ordered Field, gives

$$u(z)-u(x)\le M-u(x)\le M-m .$$

Multiplying by the nonnegative number $\lambda$, by claim 5 of Elementary Arithmetic in an Ordered Field, gives $\lambda(u(z)-u(x))\le\lambda(M-m)$, and claim 1 of Elementary Order Arithmetic in an Ordered Field combines the two displayed inequalities into

$$u(y)-u(x)\le\lambda\,(M-m).$$