Here's another proof. Assume wlog by taking converging sub-sequences and re-scaling that $a_n=0$, $\lambda_n=1/2$, and $F(0)=0$. We must show that if $b_n\to\infty$, then $D_n=\frac 1 2 F(b_n)-F(\frac 1 2 b_n)\not\to 0$. Consider the function $G(x)=\frac 1 2 F(2x)-F(x)$. By convexity, $G(x)>0$, for all $x$. Also, since $F'$ is increasing, we have $G'(x)=F'(2x)-F'(x)>0$, for all $x$. Therefore, $G$ is increasing. Therefore, $D_n=G(\frac 1 2 b_n)\not\to 0$.