Proving Threshold Properties of a Dynamic Programming Problem

Question

I'm struggling with the following simple dynamic programming problem for a couple of days. This is my first time to try to solve a dynamic programming problem analytically. The structural property is so intuitive, yet I find no way to prove it. The problem is the following: it's a finite horizon and continuous state space problem. The induction (if it should be used) is on $t$ from $t=T$ to $t=0$.