Proof of weak maximum principle for heat-type equations

Since $[0,T]$ is compact, it can be covered by a finite number of intervals $[0,\tau_{1}],[\tau_{1},\tau_{2}],\ldots, [\tau_{k-1},\tau_{k}]$, each one with length less than $1/C$ (say $1/2C$). What was done for the first interval, can be repeated in each interval as is mentioned in the proof, by the time shifting $t \mapsto t-\tau$. For example, one that you have the proof in the interval $[0,\tau_{1}]$, you work in the interval $[\tau_{1},\tau_{2}]$ and your initial conditions will be the initial conditions at time $t-\tau_{1}=0$, that is, for $t=\tau_{1}$. Since these new initial conditions have the same properties has the initial conditions when you worked in the first interval, the proof would work in this case.