**Theorem 1** Suppose \[ \dot{x}(t) \leq a(t)x(t) , \qquad t\in[0,T], \] for some nonnegative functions $x(t), a(t), b(t)$. Then \[ x(t) \leq x(0)\exp\big( \int_0^t a(s)ds \big). \] **Theorem 2** Suppose \[ \dot{x}(t) \leq a(t)x(t) + b(t), \qquad t\in[0,T], \] for some nonnegative functions $x(t), a(t), b(t)$. Then \[ x(t) \leq x(0)\exp\big( \int_0^t a(s)ds \big) + \int_0^t \exp(\int_s^t a(\tau)d\tau ) b(s)ds. \]