Proof a sufficient condition to indicate the maximal interval of existence of an IVP 's solution (Wintner 's theorem)

Question

I see this theorem was applied in an exercise book but I can't proof it.
Let an initial value problem: $$x' = f(t,x); \quad x(t_0) = x_0$$ If $f(t,x)$ is continuous on $G = [t_0,+\infty) \times(-\infty,+\infty)$ and $$ |f(t,x)| \leq a(t) |x| + b(t)$$ $a(t), b(t)$ is continuous, then the solution of this IVP is continuable on the whole of the interval $[t_0,+\infty)$

Answer 1

Suppose that $\varphi \colon [t_0,\theta) \to \mathbb{R}$, where $\theta < \infty$, is a noncontinuable solution of the IVP. One has then $$ \varphi(t) = x_0 + \int\limits_{t_0}^t f(s, \varphi(s)) \, ds \quad \forall{t \in [t_0, \theta)}, $$ hence $$ \lvert \varphi(t) \rvert \le \lvert x_0 \rvert + \int\limits_{t_0}^t \lvert f(s, \varphi(s)) \rvert \, ds \quad \forall{t \in [t_0, \theta)}. $$ It follows from the assumptions on $f$ that $$ \lvert \varphi(t) \rvert \le C + D \int\limits_{t_0}^t \lvert \varphi(s) \rvert \, ds \quad \forall{t \in [t_0, \theta)}, $$ where $$ C = \lvert x_0 \rvert + (\theta - t_0) \sup\{b(\tau): \tau \in [t_0, \theta] \}, \\ D = \sup\{ a(\tau): \tau \in [t_0, \theta] \}. $$ Grönwall's inequality gives that $$ \lvert \varphi(t) \rvert \le C e^{D (t - t_0)} \quad \forall{t \in [t_0, \theta)}, $$ which is not larger than $C e^{D (\theta - t_0)}$. We have thus obtained that the set of the values of a noncontinuable solution is bounded, and this contradicts the continuation theorem.